diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzigdr" "b/data_all_eng_slimpj/shuffled/split2/finalzzigdr"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzigdr"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction} \\label{Ss.Intro}\n\nWe consider {\\it combinatorial games} of two players;\nthey take turns alternating and one who makes the last move wins.\nBoth players have perfect information and there are no moves of chance.\nA game is called {\\it impartial} if both players\nhave the same possible moves in each position and\n\\emph{acyclic} if it is impossible to revisit the same position.\nIn this paper we consider only impartial acyclic combinatorial games\nand call them simply games.\nA more detailed introduction to combinatorial games can be found in \\cite{BCG01-04, Con76}.\n\nIf there is a move from position $x$ to $y$, we write $x \\to y$.\nFor a set $S$ of nonnegative integers the {\\it minimum excluded value} of $S$\nis defined as the smallest nonnegative integer that is not in $S$ and\nis denoted by  $\\mex (S)$. In particular, $\\mex(\\emptyset) = 0$.\nThe {\\it Sprague-Grundy} (SG) value of a position $x$\nin a game $G$ is defined recursively by  $$\\G(x) = \\mex \\{g(y) \\mid x \\to y\\}.$$\nA position of SG value $t$ is called a {\\it $t$-position};\n$0$-positions are also known as $\\P$-positions.\nA player who moves into a $\\P$-position can win the game.\nThe SG function is instrumental in the theory of disjunctive sums of games;\nsee \\cite{BCG01-04, Con76, Gru39, Spr35, Spr37}.\n\n\nA classical example is {\\sc Nim} studied by Bouton \\cite{Bou901}.\nA position in {\\sc Nim} consists of $n$ piles of tokens.\nTwo players alternately choose one of the piles and remove\nan arbitrary (positive) number of tokens from that pile.\nBouton characterized the $\\P$-positions and in fact described the SG function of {\\sc Nim}.\n\nMoore \\cite{Moo910} introduced a generalization in which\na player can remove tokens from at least one and at most  $k$  of the piles, for some fixed  $k<n$.\nWe call this game Moore's {\\sc Nim} and denote it by {\\sc Nim}$^\\leq_{n, k}$.\nWhen $k=1$ {\\sc Nim}$^\\leq_{n, 1}$ is the traditional {\\sc Nim}.\n\nMoore generalized Bouton's results and\ncharacterized the $\\P$-positions of {\\sc Nim}$^\\leq_{n, k}$ as follows.\nWe denote by $\\Z_\\geq$ the set of nonnegative integers and\nby $x \\in \\Z^n_\\geq$ a position of the game,\nwhere $x_i$ is the number of tokens in pile $i$.\nLet us represent the components of $x$ in binary form\n$x_i=\\sum_{j}x_{ij}2^j$, $i=1,...,n$, define $y_j=\\sum_{i=1}^n x_{ij} \\mod (k+1)$, and set\n\\[\nM(x)=\\sum_{j} y_j (k+1)^j.\n\\]\nFor instance, if  $n=3$, $k=2$, and $x=(2,3,6)$, then using $3$ binary digits we can write:\n$2=010$, $3=011$, and $6=110$ yielding\n$y_0=1\\mod 3=1$, $y_1=3\\mod 3=0$, and $y_2=1\\mod 3=1$,\nfrom which we get $M(x)=1\\cdot 3^0+0\\cdot 3^1+1\\cdot 3^2=10$.\n\nMoore proved that $x$ is a $\\P$-position of {\\sc Nim}$^\\leq_{n, k}$ if and only if  $M(x)=0$.\nBerge \\cite[Theorem 3, page~55]{Ber62} claimed that  $\\G(x)$ is simply equal to  $M(x)$.\nHowever, Jenkyns and Mayberry \\cite{JM80} pointed out that this is an overstatement and\nthe equality holds only when  $M(x) \\leq 1$ or $\\G(x)\\leq 1$.\nFor example, direct calculations show that\n$2 = \\G(x) < M(x) = 3$   for  $k = 2, n = 3, x = (0,0,2)$ and\n$8 = \\G(x) > M(x) = 2$   for  $k = 2, n = 3, x = (2,3,3)$.\n\nFor the case of $n = k+1$ Jenkyns and Mayberry \\cite{JM80}\nprovided a formula for the SG function of {\\sc Nim}$^\\leq_{k+1, k}$.\nAn alternative proof for a slightly more general game\nwas given recently in \\cite{BGHM15}.\n\nIn this paper we introduce another generalization of {\\sc Nim}.\nGiven positive integers $n$  and  $k$  such that $1 \\leq k \\leq n$,\nwe define {\\sc Exact $k$-Nim}, denoted by {\\sc Nim$^=_{n,k}$},  as follows.\nGiven $n$ piles of tokens, by one move a player chooses exactly $k$ piles and\nremoves arbitrary positive number of tokens from each of them.\nThe game terminates when there are less than  $k$  nonempty piles.\n{\\sc Nim}$^=_{n, k}$  turns into the standard {\\sc Nim} when  $k=1$ and\nit is the trivial one-pile {\\sc Nim} when $k=n$.\n\n\\subsection*{Main results}\n\nGiven a position $x\\in \\Z_\\geq^n$ of {\\sc Nim$^=_{n,k}$},\nwe denote by $T_{n,k}(x)$ the maximum number of consecutive moves\none can make starting with  $x$.\nWe call $T_{n,k}$ the \\emph{Tetris} function of the game.\n\nThe following two theorems characterize the SG function of {\\sc Nim$^=_{n,k}$} for $2k\\geq n$.\n\\begin{theorem} \\label{thm.n<2k}\nIf $n < 2k$ then the SG function of {\\sc Nim$^=_{n,k}$} is\nequal to its Tetris function,  $\\G(x) = T_{n,k}(x)$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm.n=2k}\nLet $k \\geq 2$, $n = 2k$, and\nlet $x = (x_1, \\ldots, x_n)$ be a position of {\\sc Nim}$^=_{2k, k}$.\nSet\n\\begin{align}\nu(x) &= T_{2k,k}(x),  \\label{eq:u}\\\\\nm(x) &= \\min_{1\\leq i\\leq 2k} x_i,\\label{eq:m}\\\\\ny(x) &= T_{2k,k}\\big(x_1 - m(x), \\ldots, x_{2k} - m(x)\\big),\\label{eq:y}\\\\\nz(x) &= 1+\\binom{y(x)+1}{2}, \\text{ and}\\label{eq:z}\\\\\nv(x) &= \\big(z(x)-1\\big) +\\big[\\big(m(x)-z(x)\\big)\\mod \\big(y(x)+1\\big)\\big].\\label{eq:v}\n\\end{align}\nThen the SG function of {\\sc Nim}$^=_{2k,k}$ is given by formula\n\\begin{equation}\\label{eq-binomial;n=2k-Formula}\n\\G(x) ~=~ \\begin{cases}\nu(x), & \\text{if~~}  m(x) < z(x);\\\\\nv(x), & \\text{if~~}  m(x) \\geq z(x).\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\nNote that this formula fails for  $k = 1$.\nIn this case {\\sc Nim}$^=_{2, 1}$ is the standard $2$-pile {\\sc Nim} and\nits SG function is the modulo $2$ sum of the two coordinates of a position,\nas described by Bouton.\nThis function is different from the one described by the above formula.\n\nWe also would like to remark that the above formula is surprisingly\nsimilar to the one given by Jenkyns and Mayberry\nin \\cite{JM80} for {\\sc Nim}$^\\leq_{k+1, k}$.\n\n\\bigskip\n\nThe above result implies a simple characterization of\nthe $0$- and $1$-positions of {\\sc Nim}$^=_{2k, k}$.\nA position $x = (x_1, \\ldots, x_n)$ is said to be\n\\emph{nondecreasing} if $x_1 \\leq x_2 \\leq \\cdots \\leq x_n$.\n\n\n\\begin{corollary}   \\label{cor-0,1-exact:n=2k}\nGiven a nondecreasing position $x = (x_1, \\dots, x_{2k})$ of the game {\\sc Nim}$^=_{2k, k}$,\n\\begin{enumerate} \\itemsep0em\n\\item [\\rm{(i)}] $x$  is a $\\P$-position if and only if\nmore than half of its smallest coordinates are equal, that is,  $x_1 = \\cdots = x_k = x_{k+1}$.\n\\item [\\rm{(ii)}] $x$  is a $1$-position if and only if $x_1 = \\cdots = x_{k - \\ell} = 2c$  and\n$x_{k-\\ell+1} = \\cdots = x_{k+ \\ell + 1} = 2c+1$\nfor some integer  $c \\in \\ZZ_\\geq$  and  $\\ell \\in \\{0,1, \\ldots, k-1\\}$.\n\\end{enumerate}\n\\end{corollary}\n\nBoth statements \\rm{(i)} and  \\rm{(ii)} follow from Theorem \\ref{thm.n=2k},\nbut can also be derived much simpler, directly from the definitions.\n\n\\medskip\n\nThe case of $2k < n$ looks much more difficult and it is still open.\nMoreover, we have not even been able to characterize\nthe $\\P$-positions of {\\sc Exact $k$-Nim} for $1 < k < n\/2$,\ne.g., for {\\sc Nim}$^=_{5,2}$.\n\n\\bigskip\n\nThe rest of the paper is organized as follows.\nIn Section \\ref{Ss.n<2k} we characterize the SG function of {\\sc Nim$^=_{n,k}$} for the case $2k>n$.\nIn Section \\ref{Ss.n=2k} we characterize the SG function of {\\sc Nim$^=_{n,k}$} for the case $2k=n$.\nIn Section \\ref{Ss.Moore01} we provide an alternative proof for the above stated result of Jenkyns and Mayberry \\cite{JM80}.\nFinally in Section \\ref{Ss.Tetris} we show that for a given position we can compute efficiently the corresponding SG value.\n\n\n\n\n\\section{SG function in the case of $n < 2k$} \\label{Ss.n<2k}\n\n\nFor our proof we need the following basic properties of the Tetris function.\nGiven positions $x,x'\\in\\Z^n_\\geq$ we write $x\\leq x'$ if $x_i\\leq x'_i$ holds for $i=1,...,n$.\n\n\\begin{lemma}\\label{tbasic}\nConsider two positions $x, x'\\in\\ZZ^n_\\geq$.\n\\begin{itemize}\n\\item [\\rm{(i)}] If $x'\\leq x$ then $T_{n,k}(x')\\leq T_{n,k}(x)$.\n\\item [\\rm{(ii)}] If in addition we have $\\sum_{i=1}^n(x_i-x_i')=1$, then $T_{n,k}(x)-1\\leq T_{n,k}(x')\\leq T_{n,k}(x)$.\n\\end{itemize}\n\\end{lemma}\n\\proof\nIt is immediate by the definition.\n\\qed\n\nA move in {\\sc Nim}$^=_{n, k}$ is called {\\it slow} if exactly one token is taken from each of the $k$ chosen piles.\n\n\n\n\\begin{lemma}\\label{Tetrisnotdecrease}\nConsider a position $x=(x_1, \\ldots, x_n) \\in \\Z^n_\\geq$ with some indices $i, j$ such that\n$x_i < x_j$. Let $x'=(x'_1, \\ldots, x'_n)$ be defined by\n\\begin{equation}\\label{eq-l2}\nx_l'=\n\\begin{cases}\nx_i+1, & \\text{ if } l=i, \\\\\nx_j-1, & \\text{ if } l=j, \\\\\nx_l,   & \\text{ otherwise. }\\\\\n\\end{cases}\n\\end{equation}\nThen we have $T_{n,k}(x) \\leq T_{n,k}(x')$.\n\\end{lemma}\nIn other words, the Tetris function is nondecreasing when\nwe move a token from a larger pile to a smaller one.\n\n\n\\proof\nConsider any sequence of slow moves from $x \\to \\cdots \\to x''$. If $x''_j>0$  then the same sequence of slow moves can be made from $x'$ since $x'_l \\geq x_l$ for $l \\neq j$.\n\nIf $x''_j=0$ then since $x_j > x_i$, this sequence contains a slow move reducing $x_j$ but not $x_i$.\nLet us modify this move reducing $x_i$ rather than $x_j$ and keeping all other moves of the sequence unchanged.\nThe obtained sequence has the same length and consists of slow moves from $x'$. \\qed\n\nNotice that we can generalize  Lemma \\ref{Tetrisnotdecrease}\nreplacing $\\pm 1$ in \\raf{eq-l2} by $\\pm \\Delta$ for any integral $\\Delta \\in [0,x_j-x_i]$.\n\n\n\\begin{lemma}\\label{bestslowmove}\nThe slow move that reduces the $k$ largest piles of $x$ reduces the Tetris value $T_{n,k}(x)$ by exactly one.\n\\end{lemma}\n\n\\proof\nLet $x'$ be the position obtained from $x$ by reducing the $k$ largest piles of $x$ by exactly one each. Let $x''$ be another position obtained by some slow move.\nBy applying \\raf{eq-l2} repeatedly, we can obtain $x'$ from $x''$ with $T_{n,k}(x'') \\leq T_{n,k}(x')$ by Lemma \\ref{Tetrisnotdecrease}.\nThis implies that $x'$ has the  highest Tetris value among all positions each reachable from x by a slow move. By Lemma \\ref{tbasic},\neach slow move reduces the Tetris value by at least one and there exists a slow move reducing it by exactly one.\nHence, $T_{n,k}(x')=T_{n,k}(x)-1$. \\qed\n\n\\bigskip\n\\noindent\\textbf{Proof of Theorem }\\ref{thm.n<2k}:\nThe Tetris value is the largest number of moves one can take from a position $x$,\nimplying that $\\G(x)$ is at most $T_{n,k}(x)$.\nTherefore, it is enough to show that for all integral $g$ such that\n$0 \\leq g < T_{n,k}(x)$ there exists a move $x\\to x'$ such that $T_{n,k}(x')= g$.\n\nConsider the move $x \\to x'$ that reduces the largest $k$ piles to $0$.\nFor the resulting position we have $T_{n,k}(x')=0$ because $2k>n$.\nLet us also consider the move $x \\to x''$ that reduces the $k$ largest piles each by only $1$.\nThen we have $T_{n,k}(x'')=T_{n,k}(x)-1$ according to Lemma \\ref{bestslowmove}.\nAny position between $x'$ and $x''$ is reachable form $x$. Thus, by Lemma \\ref{tbasic} the claim follows.  \\qed\n\n\\section{SG function in case of $n=2k$} \\label{Ss.n=2k}\n\nLet us consider the game {\\sc Nim}$^=_{2k, k}$, where $k\\geq 2$, and let $x$ be a position of this game.\nRecall that to $x$ we associated several parameters in \\eqref{eq:u}--\\eqref{eq:v}.\nBased on these parameters, we classify the positions into the following two types:\n\\begin{itemize} \\itemsep0em\n\\item [\\rm{(i)}]   \\emph{type I}, if $m(x)<z(x)$, and\n\\item [\\rm{(ii)}]  \\emph{type II} otherwise.\n\\end{itemize}\n\n\nWe shall need some technical lemmas for our proof. In this section, all positions belong to the game {\\sc Nim}$^=_{2k, k}$.\n\n\\begin{lemma} \\label{continuity}\nConsider a position $x$ and two moves $x\\to x'$ and $x\\to x''$ such that $x'\\geq x''$ $($componentwise$)$ and $y(x')\\geq y(x'')$.\nThen, for every integer $g$ with $y(x')\\geq g \\geq y(x'')$ there exists a move $x\\to x'''$ such that $y(x''')= g$ and $x'\\geq x'''\\geq x''$.\n\\end{lemma}\n\\proof\nNote that $x'\\geq x''$ implies that in the two moves $x\\to x'$ and $x\\to x''$ the same $k$ components are decreased.\nLet us denote by $K\\subseteq \\{1,2,...,2k\\}$ these components.\nLet us now start decreasing the components $x'_j$, $j\\in K$ one by one,\nkeeping their values always greater than the corresponding $x''_j$ values.\nAfter $\\sum_{j\\in K} x'_j-x''_j$ steps we reach $x''$.\nIn each of these steps the corresponding value $y$ can only decrease, and by at most $1$.\nHence there will be at least one such position $x'\\geq x'''\\geq x''$ with $y(x''') = g$.\nIt also follows that $x\\to x'''$ is a move, completing the proof.\n\\qed\n\n\nLet us also note that in fact $z$ defined in \\eqref{eq:z} depends uniquely on $y(x)$ and hence can be considered as a function of $y$.\nWe shall also need the following easy arithmetical facts.\n\\begin{lemma}\\label{interval}\nEvery nonnegative integer $g$ belongs to exactly one of the intervals\n$$[z(y)-1, z(y)+y-1] = \\left[\\binom{y+1}{2},\\binom{y+2}{2}-1\\right] \\textrm{ for }y\\in \\{0, 1, 2, ...\\}.$$\n\\qed\n\\end{lemma}\n\\begin{corollary}\\label{delta-epsilon}\nFor every nonnegative integer $g$, there exist unique $\\nu(g)$ and $\\eps(g)$ integer values such that\n\\[\ng ~=~ \\binom{\\nu(g)+1}{2} +\\eps(g) ~~~\\textbf{ and }~~~ 0\\leq \\eps(g)\\leq \\nu(g).\n\\]\n\\end{corollary}\n\n\\smallskip\n\nGiven a nondecreasing position $x$, let us construct another position $\\bar x$ from $x$ by emptying the first $n-k$ piles and adding these $\\sum_{i=1}^{n-k}x_i$ tokens, one by one, to the last $k$ piles as follows:\nIn each step we add one token to the smallest of these $k$ piles.\nIf there are several such piles, we break the tie by adding this token to the pile of the largest index.\n\nIt is easy to see that we have $T_{n,k}(\\bar x)=\\min(\\bar x_i  |  n-k+1 \\leq i \\leq n)=\\bar x_{n-k+1}$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=6cm]{Doc1}\n\\caption{$k=4,n=7$, $x=(1,2,2,3,4,4,7)$, and $\\bar x=(0,0,0,5,5,6,7)$.} \\label{ADD(1)}\n\\end{figure}\n\n\n\n\\begin{lemma}\\label{txequalstxbar}\nThe above construction of $\\bar x$ from $x$ keeps the Tetris value unchanged: $T_{n,k}(\\bar x)=T_{n,k}(x)$.\n\\end{lemma}\n\\proof\nLet us note that $T_{n,k}(\\bar x) \\leq T_{n,k}(x)$ by Lemma \\ref{Tetrisnotdecrease}.\nBy the above definition of $\\bar x$, none of the tokens from the smallest $n-k$ piles of $x$ are moved to any pile of size larger than $T_{n,k}(\\bar x)+1$ and hence we have\n\\begin{equation}\\label{volumeargument}\nV(x):=\\sum_{i=1}^n \\min(x_i,T_{n,k}(\\bar x)+1) =\\sum_{i=n-k+1}^n \\min(\\bar x_i,T_{n,k}(\\bar x)+1).\n\\end{equation}\nSince $\\bar x_{n-k+1}=T_{n,k}(\\bar x)$, we get by \\eqref{volumeargument} that\n\\begin{equation*}\nV(x)  \\leq k-1+ k T_{n,k}(\\bar x) < k(T_{n,k}(\\bar x)+1).\n\\end{equation*}\nAssume now indirectly that $T_{n,k}(\\bar x) < T_{n,k}(x)$.\nThen it is possible to construct a  sequence of $T_{n,k}(\\bar x)+1$ slow moves from $x$.\nBy such sequence any pile would be reduced at most $T_{n,k}(\\bar x)+1$ times,\nand therefore the total number of the removed tokens is at most $V(x)$,\nimplying $k(T_{n,k}(\\bar x)+1)\\leq V(x)$, contradicting the above inequality.\nThe obtained contradiction implies that $T_{n,k}(x)=T_{n,k}(\\bar x)$. \\qed\n\n\\begin{lemma}\\label{getmeTetris}\nLet $x = (x_1, \\ldots, x_n)$ be a position.\nIf there exists a nonnegative integer $g$ such that\n\\begin{equation}\\label{eq-getmeTetris}\nk g \\leq \\sum_{i=1}^{n} \\min(x_i,g) \\quad \\text{ and } \\quad \\sum_{i=1}^{n} \\min(x_i,g+1) < k(g+1),\n\\end{equation} \\label{E.delta}\nthen we have $T_{n,k}(x)=g$.\n\\end{lemma}\n\\proof\nThe sequence $s(g)=\\frac{1}{g}\\sum_{i=1}^n \\min(x_i,g)$ is monotone non-increasing for $g \\in \\ZZ_\\geq$,\nand hence, the inequalities \\raf{eq-getmeTetris} can hold for at most one $g$.\nWithout any loss of generality we can assume that $x$ is nondecreasing, and thus we can define $\\bar x$, as above.\nLemma \\ref{txequalstxbar} then implies that \\raf{eq-getmeTetris} holds for $g=T_{n,k}(x)=T_{n,k}(\\bar x)$. \\qed\n\n\\bigskip\n\n\\noindent{\\bf Proof of Theorem \\ref{thm.n=2k}.}\nOur main result claims that the SG function $\\G(x)$ of\n{\\sc Nim}$^=_{2k, k}$ for $k\\geq 2$ is equal to the function\n\n$$\ng(x)=\\begin{cases}\nu(x), & \\text{if }m(x) < z(x); \\\\\nv(x), & \\text{if }m(x) \\geq z(x);\n\\end{cases}\n$$\nwhere $ u,m,y,v,$ and $z$ are defined in \\eqref{eq:u}--\\eqref{eq:v}.\nTo prove this theorem, it is enough to show the following two properties of $g(x)$:\n\\begin{itemize}\n\\item[(I)] for any move $x \\rightarrow x'$ we have $g(x) \\neq g(x')$ and\n\\item[(II)] for every value $\\delta$ such that $0 \\leq \\delta < g(x)$ there exists a move $x \\to x'$ with $g(x')=\\delta$.\n\\end{itemize}\n\nWe assume that $x$ is nondecreasing.\n\n\\subsection{Proof of (I)}\n\nWe now prove property (I). Let us first consider the case $z(x) > m(x)$.\nObviously, any move $x \\rightarrow x'$ reduces the Tetris value by at least 1, implying $u(x) > u(x')$.\nUsing this and the definitions, we get $g(x)=u(x) > u(x')$. If $g(x') = u(x')$ then by the above inequality, we get $g(x) \\neq g(x')$.\nOn the other hand, if $g(x') = v(x')$ then we  must have $m(x') \\geq z(x')$ and thus $v(x')=z(x')-1 + \\left((m(x')-z(x')) \\mod (y(x')+1)\\right)\\leq z(x')-1 + (m(x')-z(x'))=m(x')-1$.\nThus, we get $g(x) = u(x) > u(x') > m(x')-1 \\geq  v(x') = g(x')$.\n\nIt remains to consider the case $z(x) \\leq m(x)$, in which case $v(x)\\leq m(x)-1$ follows by the definitions. \n\t\\begin{enumerate} \\itemsep0em\n\t\\item Suppose $z(x') > m(x')$.\n    We can estimate $u(x') \\geq m(x)+m(x')$ since $2k = n$.\n    Furthermore, $g(x)=v(x) \\leq m(x)-1$ and thus $g(x')=u(x') \\geq m(x)+m(x') > m(x)-1 \\geq g(x)$.\n    \\item Suppose $z(x') \\leq m(x')$. Then $g(x')=v(x')$.  Note that $z(x) \\leq m(x)$ and thus $g(x)=v(x)$.\n    Also note that $m(x') \\leq m(x)$. We examine the last inequality.\n    \t\\begin{enumerate} \\itemsep0em\n\t    \\item Suppose $m(x')=m(x)$.\n               \n                Then $x-m(x)\\to x'-m(x')$ is a move, and hence decreases the Tetris value implying $y(x)>y(x')$.\n\t           By Lemma \\ref{interval}, we have\n                $z(x')+y(x')-1 < z(x)-1$\n                and so\n                $z(x)  > z(x') + y(x')$\n                implying\n                $v(x) \\geq z(x)-1> z(x')-1+y(x') \\geq v(x')$,\n                since\n                $y(x') \\geq \\bigg(\\big(m(x') - z(x)\\big) \\mod\\big(y(x) + 1\\big)\\bigg)$,\n                regardless  of the value of $m(x') - z(x)$.\n\t    \\item Suppose $m(x') < m(x)$. We compare $y(x)$ with $y(x')$.\n            \\begin{enumerate} \\itemsep0em\n\t        \\item If $y(x)= y(x')$ then $z(x)=z(x')$. By the definition of a legal move, $x'$ has at least $k$ piles not smaller than $m(x)$. Therefore, $1 \\leq\n            m(x)-m(x') \\leq y(x')=y(x)$, which implies that $m(x) \\mod (y(x)+1) \\neq m(x') \\mod (y(x)+1)$ and thus $v(x) \\neq v(x')$.\n\t        \\item If $y(x) \\neq y(x')$, since the $y$ values are different, by Lemma \\ref{interval} we have that $v(x)$ and $v(x')$ are in different intervals,\n            therefore $v(x)\\neq v(x')$.\n            \\end{enumerate}\n\t   \\end{enumerate}\n\t\\end{enumerate}\n\n\\subsection{Proof of (II)}\n\nWe prove property (II) by considering type I and type II positions, separately.\n\n\\subsubsection{Type I positions: $m(x)<z(x)$}\n\nFirst, let us consider the case $m(x)=0$. Then there are at most $2k-1$ nonempty piles.\nSo we can reduce the Tetris value $T_{2k,k}(x)$ to $0$ by emptying the $k$ largest piles of $x$.\nTherefore, by Lemmas \\ref{achieveTetrisvalue} and \\ref{continuity}, for any $0 \\leq \\delta < T_{2k,k}(x)$, there exists a move $x \\to x'$ with $T_{2k,k}(x')=\\delta$.\nAll such moves also have $m(x')=0$, therefore $m(x')<z(x')$.\n\nFrom now on we can assume that $m(x) \\geq 1$. Let us consider the following four subcases, depending on the value $\\delta$.\n\n\\begin{enumerate}\n\\item $0\\leq \\delta <  m(x)$.\nTo simplify our proof, let us use simply $m$ instead of $m(x)$ in this section.\nLet us observe first that since $x$ is type I we have $m<z(x)$ implying by \\eqref{eq:z} that\n\\begin{equation}\\label{sqrt-m1}\ny(x)>\\nu(m-1)\\geq 0.\n\\end{equation}\nwith $\\nu$ being defined as in Corollary \\ref{delta-epsilon}.\n\nLet us next define a set $Q=Q(x)$ of pairs of integers by setting\n\\[\nQ^1(x)=\\left\\{(\\mu,\\ge)\\left| \\begin{array}{c}m-\\ge~\\leq~ \\mu~\\leq~ m\\\\\n\\ge ~\\leq \\nu(m-1)-1\n\\end{array}\\right.\\right\\}\n\\]\n\\[\nQ^2(x)=\\left\\{(\\mu,\\ge) \\left| \\begin{array}{c}m-\\eps(m-1)~\\leq~ \\mu~\\leq~ m\\\\\n\\ge ~= \\nu(m-1)\n\\end{array}\\right. \\right\\} ,\n\\]\nand defining $Q=Q^1(x)\\cup Q^2(x)$.\n\nWe show that if for a position $x^*$ we have $(m(x^*),y(x^*))\\in Q$, then $x^*$ is of type II.\nTo see this consider a pair $(\\mu,\\ge)\\in Q^1(x)$.\nThen we have by the definition of $Q^1(x)$ that $\\mu\\geq m-\\ge$ and that $\\nu(m-1)-1\\geq \\ge$ from which $z(\\nu(m-1)-1)\\geq z(\\ge)$ follows by the definition of $z$ in \\eqref{eq:z}.\nWe also have the inequality $m-\\nu(m-1)+1\\geq z(\\nu(m-1)-1)$ since $m\\geq \\binom{\\nu(m-1)+1}{2}$ by the definition of $\\nu$ in Corollary \\ref{delta-epsilon}.\nPutting these together, we obtain $\\mu\\geq z(\\ge)$ as stated.\nFor $(\\mu,\\ge)\\in Q^2(x)$ we have $\\mu\\geq m-\\epsilon(m-1)$ and $\\ge=\\nu(m-1)$ by the definition of $Q^2(x)$.\nSince $m-1=\\binom{\\nu(m-1)+1}{2}+\\epsilon(m-1)$ by Corollary \\ref{delta-epsilon},\nthe inequality $\\mu\\geq m-\\epsilon(m-1)=z(\\ge)$ follows again.\n\nWe show next that for all pairs $(\\mu,\\ge)\\in Q$ there exists a move $x\\to x^*$ such that $\\mu=m(x^*)$ and $\\ge=y(x^*)$ (and $x^*$ is of type II, as we argued in the previous paragraph.)\nFor this let us consider first $\\mu\\leq m-1$ and note that if $(\\mu,\\ge)\\in Q$ then $m-\\mu\\leq \\ge\\leq \\nu(m-1)$.\nOur plan is to use Lemma \\ref{continuity} and to cover this range of $\\ge$ values by two constructions.\n\nLet us define a pair of positions $x'\\geq x''$ by\n\\begin{align*}\nx'_i &=\n\\begin{cases}\nx_i,   &\\text{ for } i=1 \\text{ and } i\\geq k+2,\\\\\n\\mu,   &\\text{ for } i=2,                    \\\\\nx_i-1, &\\text{ for } i=3,...,k+1\n\\end{cases} \\\\\n\\intertext{and}\nx''_i &=\n\\begin{cases}\nx_i,  &\\text{ for } i=1 \\text{ and } i\\geq k+2,\\\\\n\\mu,  &\\text{ for } i=2,...,k+1.\n\\end{cases}\n\\end{align*}\nNote that since $\\mu<m=x_1\\leq x_2\\leq x_i$ for $i\\geq 3$, both $x'$ and $x''$ are reachable from $x$. We claim next that $y(x'')=m-\\mu$ and $y(x')\\geq \\nu(m-1)$.\nThe first claim follows easily, since in $x''$ we have exactly $k$ positions larger than $\\mu$, and $x_1=m=\\mu +(m-\\mu)$.\nFor the second let us note that since $x_1-\\mu=m-\\mu\\geq 1$ we have $y(x')\\geq x_{k+1}-\\mu$. We can now apply Lemma \\ref{continuity} for $x'$ and $x''$,\nand conclude that for all values $m-\\mu \\leq \\ge \\leq x_{k+1}-\\mu\\leq y(x')$ there exists a move $x\\to x^*$ such that $m(x^*)=\\mu$ and $y(x^*)=\\ge$.\nFor larger values of $\\ge$ we need a modified construction:\n\\begin{align*}\nx'_i & =\n\\begin{cases}\n\\mu&\\text{ for } i=1,\\\\\nx_i-1, &\\text{ for } i=2,...,k,\\\\\nx_i,   &\\text{ for } i=k+1,...,2k,\n\\end{cases}\\\\\n\\intertext{and}\nx''_i & = \\begin{cases}\n\\mu,   &\\text{ for } i=1,...,k,\\\\\nx_i,   &\\text{ for } i=k+1,...,2k.\n\\end{cases}\n\\end{align*}\nAssuming $\\mu < m$, we have $y(x')\\geq y(x)\\geq \\nu(m)\\geq \\nu(m-1)$,\nwhile $y(x'')=x_{k+1}-\\mu$. We have again $m(x')=m(x'')=\\mu$. Thus by applying\nLemma \\ref{continuity} for $x'$ and $x''$ we can conclude that for all values $x_{k+1}-\\mu\\leq \\ge \\leq \\nu(m-1)$ there exists a move $x\\to x^*$ such that $m(x^*)=\\mu$ and $y(x^*)=\\ge$.\n\n\\medskip\n\nFinally, for $\\mu=m$ we proceed analogously, but with a third construction. Note first that $(m,\\ge)\\in Q$ if and only if $0\\leq \\ge\\leq \\nu(m-1)$.\nLet us now proceed with constructing two positions reachable form $x$:\n\\begin{align*}\nx'_i &=\n\\begin{cases}\nx_i,  &\\text{ for } i=1,...,k,\\\\\nx_i-1, &\\text{ for } i=k+1,...,2k\n\\end{cases} \\\\\n\\intertext{and}\nx''_i &=\n\\begin{cases}\nx_i,   &\\text{ for } i=1,...,k,\\\\\nm,     &\\text{ for } i=k+1,...,2k.\n\\end{cases}\n\\end{align*}\nIt is easy to see that both are reachable from $x$, and that $m(x')=m(x'')=m$, $y(x')\\geq y(x)-1\\geq \\nu(m-1)$ by \\eqref{sqrt-m1} and $y(x'')=0$.\nThus, the existence of an $x^*$ reachable from $x$ with $m(x^*)=m$, $y(x^*)=\\ge$ follows by Lemma \\ref{continuity} for all $0\\leq \\ge\\leq \\nu(m-1)$.\n\nBy the above arguments we have a move $x\\to x^*$ to a type II position $x^*$ with $m(x^*)=\\mu$ and $y(x^*)=\\ge$ for all $(\\mu,\\ge)\\in Q$.\nTo conclude the proof for this case we claim that the corresponding $v(x^*)$ values include all integers in the interval $[0,m-1]$.\nNote that $v(x^*)$ depends only on $m(x^*)$ and $y(x^*)$ for type II positions by \\eqref{eq:v},\nand that for a fixed value of $\\ge\\leq \\nu(m-1)-1$ we have exactly $\\ge+1$ consecutive integer values for $\\mu$ such that $(\\mu,\\ge)\\in Q$, implying that the corresponding values\n$v(\\mu,\\ge)=\\big(z(\\ge)-1\\big) + \\big[\\big(\\mu-z(\\ge)\\big)\\mod (\\ge+1)\\big]$\nis exactly the set of integers in the interval\n$[z(\\ge)-1, z(\\ge)+ \\ge-1]$.\nThus our claim follows by the construction of $Q$ and by Lemma \\ref{interval}.\n\n\\bigskip\n\n\\item $m(x) \\leq \\delta < x_2$ (Figure \\ref{F1}).\nSet $x'_i=0$ for $i=k+1, \\ldots ,n-1$ and $x'_n=\\delta-x_1$.\n\nBy definition $x_n\\geq x_2>\\delta\\geq x_1$ which implies $x'_n=\\delta-x_1<x_n-x_1\\leq x_n$.\nFurthermore, we have $x_i\\geq x_2>0$ for $i=k+1,...,n-1$,\nand therefore we indeed decrease exactly $k$ piles of $x$ to obtain $x'$. Thus $x'$ is reachable from $x$.\n\nFor $x'$ we have\n\\begin{align*}\n\\sum_{i=1}^{2k}\\min(x_i',\\delta)   &\\geq x_1' + (k-1)\\delta + (\\delta-x_1') =k\\delta \\\\\n\\intertext{and}\n\\sum_{i=1}^{2k}\\min(x_i',\\delta+1) &\\leq x_1' + (k-1)(\\delta+1) + (\\delta - x_1)\\\\\n                                   &= \\delta + (k-1)(\\delta +1) < k(\\delta +1).\n\\end{align*}\nTherefore, by Lemma \\ref{getmeTetris}, we have $T_{2k,k}(x')=\\delta$.\n\nSince $k\\geq 2$, $x'_{k+1}=0$ and therefore $m(x')=0$. Thus, $m(x')=0<1\\leq z(x')$ implying\nthat $x'$ is of type I, from which $g(x')=u(x')=\\delta$ follows by the above.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=7cm]{Doc2}\n\\caption{$x'$ is obtained by removing the gray area.} \\label{F1}\n\\end{figure}\n\n\\item $x_2 \\leq \\delta  < u(x)-m(x)$ (Figure \\ref{F2}).  Let $A=\\sum_{i=2}^{k+1} \\max(0,\\delta-x_i)$.\nFor $i=k+2, \\ldots ,n$ choose $a_i$ such that $0 \\leq a_i \\leq \\min(x_i-1,\\delta)$ and $\\sum_{i=k+2}^n a_i=A$.\nFirst let us prove that this is possible, or equivalently that $A \\leq \\sum_{i=k+2}^n \\min(\\delta,x_i-1)$.\nTo see this let us consider two cases.\nIf $x_{k+1}>\\delta$ then $\\sum_{i=k+2}^n \\min(\\delta,x_i-1)\\geq (k-1)\\delta \\geq A$.\nIf $x_{k+1}\\leq \\delta$ then let us define\n\\begin{equation} \\label{B.def}\nB=\\sum_{i=2}^{k+1}x_i = k\\delta - A,\n\\end{equation}\nand observe that for any integer $t\\geq\\delta >x_{k+1}$ we have\n\\[\n\\sum_{i=1}^n\\min(t,x_i) = m + B +\\sum_{i=k+2}^n\\min(t,x_i).\n\\]\nConsequently, since we have $T_{2k,k}(x)=u(x)$, by Lemma \\ref{getmeTetris} we can write for $t=u(x)$ that\n\\[\nkt \\leq \\sum_{i=1}^n\\min(t,x_i) = m(x)+B+\\sum_{i=k+2}^n\\min(t,x_i).\n\\]\nNote that if we decrease $t=u(x)$ by $1$,\nthen the left hand side decreases by $k$ while the right hand side decreases by at most $k-1$,\nhence the inequality remains valid. Let us repeat this $m(x)$ times, obtaining the inequality\n\\[\nk(u(x)-m(x))+km(x) \\leq m(x)+B+\\sum_{i=k+2}^n\\min(u(x)-m(x),x_i)+(k-1)m(x)\n\\]\nfrom which\n\\[\nk(u(x)-m(x))\\leq B+\\sum_{i=k+2}^n\\min(u(x)-m(x),x_i)\n\\]\nfollows. Let us now decrease $t=u(x)-m(x)$ further by $1$, as well as replace $x_i$ by $x_i-1$.\nThen the left hand side decreases by exactly $k$, while the right hand side decreases by at most $k$, yielding the valid inequality\n\\[\nk(u(x)-m(x)-1)\\leq B+\\sum_{i=k+2}^n\\min(u(x)-m(x)-1,x_i-1).\n\\]\nFinally, we can decrease $t=u(x)-m(x)-1$ further on both sides to $t=\\delta$ and similarly to the above argument obtain\n\\[\nk\\delta\\leq B+\\sum_{i=k+2}^n\\min(\\delta,x_i-1).\n\\]\nBy \\eqref{B.def} we obtain the claimed inequality, and hence the proof for the existence of the $a_i$ values for $k=k+2,...,n$\nthat satisfy the desired inequalities.\n\nLet us now consider the position $x'$ defined by\n\\[\nx'_i =\n\\begin{cases}\n0,     & \\textrm{ for } i=1; \\\\\nx_i,   &\\textrm{ for }i=2, \\ldots, k+1; \\\\\na_i,   &\\textrm{ for }i=k+2, \\ldots ,n.\n\\end{cases}\n\\]\nBy the above arguments $x\\to x'$ is a move in the game.\nThe equality $g(x')=u(x')=\\delta$ now follows by the above analysis and Lemma \\ref{getmeTetris}, completing our proof in this case.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=9cm]{Doc3}\n\\caption{$x'$ is obtained by removing the gray area.} \\label{F2}\n\\end{figure}\n\n\n\\item $u(x)-m(x) \\leq \\delta < u(x)$ (Figure \\ref{F3}).\nLet us define position $x'$ as follow\n$$\nx_i' =\n\\begin{cases}\nx_i-u(x)+\\delta,  &\\textrm{ for } i \\in I_1=\\{i \\mid i\\leq k, x_{i+k} \\leq \\delta\\};   \\\\\nx_i-u(x)+\\delta,  &\\textrm{ for } i \\in I_2=\\{i \\mid i > k,   \\delta < x_i \\leq u(x)\\}; \\\\\n\\delta ,           &\\textrm{ for }i \\in I_3=\\{i \\mid  i > k,   x_i > u(x) \\}; \\\\\nx_i,              &\\textrm{ otherwise.}\n\\end{cases}\n$$\nNote that $x_i \\geq m(x)\\geq u(x)-\\delta$, therefore $x_i'$ are all nonnegative.\nIt is easy to see that $I_1+k,I_2,I_3$ form a partition of $\\{ k+1, \\ldots , n \\}$.\nWe have reduced $x_i$ for all $i \\in I_1 \\cup I_2 \\cup I_3$, therefore $x\\to x'$ is a move.\n\nNext we note that the above construction implies that\n\\[\n\\sum_{i=1}^n \\min(x_i',\\delta) \\geq \\sum_{i=1}^n\\min(x_i,u(x)) - k (u(x)-\\delta)\\geq k \\delta\n\\]\nwhere the last inequality follows by the fact that $u(x)=T_{2k,k}(x)$, and hence $\\sum_{i=1}^n\\min(x_i,u(x))\\geq ku(x)$ by Lemma \\ref{getmeTetris}.\nSimilarly, we get\n\\[\n\\sum_{i=1}^n \\min(x_i',\\delta+1) \\leq \\sum_{i=1}^n\\min(x_i,u(x)+1) - k (u(x)-\\delta)< k (\\delta+1)\n\\]\nsince $\\sum_{i=1}^n\\min(x_i,u(x)+1)< k(u(x)+1)$.\nTherefore $T_{n,k}(x')=u(x')= \\delta$ follows by Lemma \\ref{getmeTetris}.\n\nNote that $x_{k+1}\\leq \\delta$, since otherwise $u(x)\\geq \\delta+m(x)$ would follow, contradicting our choice of $\\delta$.\nIt follows that $x_1 \\in I_1$ and thus $m(x')=m(x)-u(x)+\\delta<m(x)$.\n\nLet us define $\\hat{x}_i=\\min(x_i,u(x))$ for $i=1,...,n$. Then, by Lemma \\ref{Tetrisnotdecrease}, we have $y(x)=y(\\hat{x})$ and by construction of $x'$\nwe have $\\hat{x}_i-m(x)\\leq x'_i-m(x')$ for all indices $i$, implying $y(x)=y(\\hat{x}) \\leq y(x')$ by Lemma \\ref{tbasic}.\n\nFinally, $m(x')=m(x)-u(x)+\\delta<m(x)<z(x)\\leq z(x')$, which implies $g(x')=u(x')=\\delta$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=10.5cm]{Doc4}\n\\caption{$x'$ is obtained by removing the gray area.} \\label{F3}\n\\end{figure}\n\n\n\\end{enumerate}\n\n\\subsubsection{Type II positions: $m(x)\\geq z(x)$}\n\n\nFor a position $x$ let\n\\begin{align}\nD_1(x) &=\\{(m,y)\\mid 0\\leq m\\leq m(x) , m(x)-m\\leq y< y(x) \\},\\\\\nD_2(x) &=\\{(m,y)\\mid m(x)-v(x)+z(x)-1\\leq m <m(x) , y=y(x) \\},\\\\\nD(x) &=D_1(x)\\cup D_2(x).\n\\end{align}\nNote that $D_2(x)=\\emptyset$ whenever $(m(x)-z(x))\\mod (y(x)+1)$ is zero.\n\n\\begin{lemma}\\label{existsmoveinD}\nFor every $x$ and for every $(m,y) \\in D(x)$ there is a position $x'$ reachable from $x$ such that $m(x')=m$ and $y(x')=y$.\n\\end{lemma}\n\n\\proof\nFor $(m,y) \\in D(x)$ let us consider the following three cases.\n\nCase 1:\n$0 \\leq m< m(x)$ and $x_2-m \\leq y \\leq y(x)$.\nWe consider two positions $\\bar{x}$ and $\\hat{x}$ reachable from $x$ defined as follows\n$$\\bar{x}_i =\n\\begin{cases}\nm,    & \\textrm{ if either } i=1 \\textrm{ or } i\\geq k+2;\\\\\nx_i,  & \\textrm{ otherwise};\n\\end{cases}\n$$\n$$\n\\hat{x}_i =\n\\begin{cases}\nx_i,                    & \\textrm{ if } i \\leq k+1; \\\\\n\\min(x_i,m(x)+y(x))-1,  & \\textrm{ if } i \\geq k+2;\n\\end{cases}\n$$\nSince we have $y(\\bar{x})=x_2-m$ and $y(\\hat{x})\\geq y(x)-1+m(x)-m \\geq y(x)$,\nwe can apply Lemma \\ref{continuity} and obtain that\nfor all values $x_2-m\\leq y\\leq y(x)$ there exists a position $x'$ such that\n$\\bar{x}\\leq x'\\leq \\hat{x}$ and $y(x')=y$. All these positions have $m(x')=m$.\n\nLet us also note that all these positions $x'$\nare reachable from $x$, because exactly $k$ components of $x$ are decreased.\n\n\\medskip\n\nCase 2: $0\\leq m < m(x)$ and $m(x)-m \\leq y < x_2-m$.\n\nWe consider two positions $\\bar{x}$ and $\\hat{x}$ reachable from $x$ defined as\n$$\n\\bar{x}_i =\n\\begin{cases}\nm,   & \\textrm{ if } i=2 \\textrm{ or } i\\geq k+2;\\\\\nx_i, & \\textrm{ otherwise;}\n\\end{cases}\n$$\n$$\\hat{x}_i =\n\\begin{cases}\nm,   & \\textrm{ if } i=2;\\\\\nx_i, & \\textrm{ if } i=1 \\textrm{ or } i=3,4, \\ldots, k+1; \\\\\n\\min(x_i,m(x)+y(x))-1, & \\textrm{ if } i \\geq k+2.\n\\end{cases}\n$$\n\nSince we have $y(\\bar{x})=m(x)-m$ and\n$y( \\hat{x}) \\geq x_{k+1}-m-1 \\geq x_2-m-1$, we can apply Lemma \\ref{continuity}, and obtain\nthat for all $y \\in[m(x)-m,x_2-m-1]$ there exists a position $x' \\in [\\bar{x},\\hat{x}]$ such that $y=y(x')$.\n\n\\medskip\n\nCase 3: $m=m(x)$ and $0 \\leq y < y(x)$.\nLet us note that the last inequaliti implies $y(x) \\geq 1$ and therefore $x_{k+1}>m(x)$.\nWe consider two positions $\\bar{x}$ and $\\hat{x}$ reachable from $x$ defined as\n$$\\bar{x}_i =\n\\begin{cases}\nx_i,   & \\textrm{ if } i=1, \\ldots ,k;\\\\\nm(x),  & \\textrm{ if } i\\geq k+1; \\\\\n\\end{cases}\n~~~~\\hat{x}_i =\n\\begin{cases}\nx_i,   & \\textrm{ if } i=1, \\ldots ,k; \\\\\nx_i-1, & \\textrm{ if } i\\geq k+1. \\\\\n\\end{cases}\n$$\nSimilarly to the previous cases, we have $y(\\bar{x})=0$ and\n$y(\\hat{x}) = y(x)-1$, and thus by Lemma \\ref{continuity} it follows that\nfor all $y \\in[0,y(x)-1]$ there exists an $x' \\in [\\bar{x},\\hat{x}]$ such that $y=y(x')$.\n\nIf we put all three cases together we cover all values $(m,y) \\in D$.\n\\qed\n\n\\medskip\n\nLet us set\n\\begin{equation} \\label{v(m,y)}\nv(m,y)= {y+1 \\choose 2}+ \\Big[ \\Big(m-1-{y+1 \\choose 2}\\Big)\\mod (y+1)\\Big].\n\\end{equation}\nNote that if $m=m(x)$ and $y=y(x)$ then $v(m,y)=v(x)$.\nFurthermore, we have\n$$V(y):=\\{v(m,y) | m \\in \\ZZ_\\geq\\}=\\left[ {y+1 \\choose 2},{y+2 \\choose 2} \\right).$$\nTherefore the sets $V(y)$, $y \\in \\ZZ_\\geq$ form a partition of $\\ZZ_\\geq$, as shown in Lemma \\ref{interval}.\n\n\\begin{lemma}\\label{Dstar}\nIf $m(x) \\geq z(x)$, then every $(m,y) \\in D(x)$ satisfies the following relations\n$$m\\geq {y+1 \\choose 2}+1~~\\textrm{ and }~~\\{v(m,y) \\mid (m,y) \\in D(x) \\} = [0,v(x)).$$\n\\end{lemma}\n\n\\proof\nLet us first consider $(m,y) \\in D_1(x)$.\nBy the definition of $D_1$ and the assumption of  $m(x) \\geq z(x)={y(x)+1 \\choose 2}+1$ we get\n$$m\\geq m(x)-y \\geq {y(x)+1 \\choose 2}+1-y \\geq {y+2 \\choose 2}+1-y \\geq {y+1 \\choose 2}+1.$$\nBy the definition of $D_1(x)$, for all $y \\in [0,y(x))$ we have $(m,y) \\in D_1(x)$ for all $m \\in [m(x)-y,m(x)]$. Hence, by (\\ref{v(m,y)}) we have $\\{v(m,y) \\mid m \\in [m(x)-y,m(x)]\\}=V(y)$.\nThus, by Lemma~\\ref{interval} we get $$\\{v(m,y) \\mid (m,y) \\in D_1(x) \\}=\\bigcup_{y \\in [0,y(x))}V(y)=[0,z(x)-1).$$\n\n\\smallskip\n\nLet us next consider $(m,y) \\in D_2(x)$.\nBy the definition of $ v(x)$ we can write $v(x)=z(x)-1+r $, where $r=m(x)-z(x)-\\lambda(y(x)+1)$ for some $\\lambda \\in \\ZZ_\\geq$. This implies $m(x)-r\\geq z(x)$. By the definition of $D_2(x)$ we have $m\\geq m(x)-r$.\nThese two inequalities imply $m\\geq {y+1 \\choose 2} +1$.\nSince $m \\in [m(x)-r, m(x))$ takes $r$ consecutive values, we have\n$\\{v(m,y) \\mid (m,y) \\in D_2(x) \\}=[z(x)-1,v(x))$.\n\\qed\n\nThe above lemma implies that any position $x'$ with $m(x')=m$, $y(x')=y$ for some $(m,y)\\in D$\nis a type II position. Hence $g(x')=v(x')=v(m,y)$.\nThus the second claim in the above lemma together with Lemma \\ref{existsmoveinD} implies that for any $0\\leq \\delta <v(x)$ there exists a move $x\\to x'$ such that $g(x')=\\delta$.\n\nSince we proved this for both type I and type II positions we concluded the proof of (II).\n\nProperties (I) and (II) together now imply that $\\G=g$.\nThis concludes the proof of Theorem \\ref{thm.n=2k}.\n\\qed\n\n\\smallskip\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=7cm]{Doc5}\n\\caption{Case $n=6,k=3$. Both above positions have $m=1,u=7,y=5$; but the position obtained by removing the gray area has $m=1,u=2,y=0$ and such a position cannot be obtained by the picture on the right since $m=1$ implies that $u \\geq 4$.} \\label{n=6,k=3}\n\\end{figure}\n\n\\section{Characterization of the $0$- and $1$-positions in Moore's game} \\label{Ss.Moore01}\n\nThe following result was shown in \\cite{JM80}. We provide here a different proof for the convenience of the reader.\n\n\\begin{theorem} $($see \\cite{JM80}$)$. \\label{t-Moore-m=0,1}\nFor any $k \\in \\{1, \\ldots, n\\}$  and  $m \\in \\{0,1\\}$, a position\n$x$  is an $m$-position of the Moore game  {\\sc Nim}$^\\leq_{n, k}$\nif and only if  $M(x) = m$.\n\\end{theorem}\n\nLet us represent the components of $x$ as binary sequences\n\\[\nx_i=\\sum_{j=0}^\\infty x_{ij}2^j \\text{ for } i=1,...,n,\n\\]\nand define\n\\begin{equation}\\label{e0001}\ny_j=\\sum_{i=1}^n x_{ij} \\mod (k+1) \\text{ for } j=0,1,....\n\\end{equation}\nThen we have $M(x)=\\sum_{j=0}^\\infty y_j(k+1)^j$.\n\n\\begin{lemma}\\label{moore1}\nLet  $x\\to x'$ be a move, and let $j$ be the highest index such that $x_{ij}\\neq x'_{ij}$ for some $i$.\nThen we must have $x'_{ij}\\leq x_{ij}$ for all $i=1,...,n$.\n\\end{lemma}\n\\proof\nIn a move we can only decrease the components of $x$.\nTherefore, if $x'_{ij}>x_{ij}$, then we must have a $j'>j$ such that $x'_{ij'}<x_{ij'}$.\n\\qed\n\n\n\\subsection{Proof of Theorem \\ref{t-Moore-m=0,1} for $m=0$}   \\label{ss-p-t1-0}\n\nFirst, let us prove Moore's result:\n$\\cG(x)=0$ if and only if  $M(x)=0$.\n\n\\smallskip\n\nBy the properties of $\\P$-positions it is enough to show that\n\\begin{itemize}\n\\item [\\rm{(i0)}] for any position $x$ with $M(x)=0$, there exists no  move $x \\to x'$ such that $M(x') = 0$;\n\\item [\\rm{(a0)}] for any position $x$ with $M(x)> 0$,  there exists a move $x \\to x'$ such that  $M(x') = 0$.\n\\end{itemize}\n\nTo show \\rm{(i0)}, let us consider a move $x \\to x'$\nfrom a position  $x$  with  $M(x)=0$.\n\nLet  $j$  be the highest binary bit such that\n$x_{ij}$ and $x'_{ij}$  differ for some $i$. Such a $j$ must exist since in a move we must change at least one components.\nBy Lemma \\ref{moore1} we have $x'_{ij}\\leq x_{ij}$ for all $i=1,...,n$,\nimplying $1\\leq \\sum_{i=1}^n(x_{ij}-x'_{ij}) \\leq k$ because  in a move we can change at most $k$ components. Therefore,\n$(\\sum_{i}x'_{ij} \\mod (k+1)) \\not= 0$ and, thus,  $M(x') > 0$.\n\n\\medskip\n\nTo show \\rm{(a0)}, let us consider a position  $x$  with  $M(x)>0$. We will construct a move $x \\to x'$ such that $M(x') = 0$.\n\n\\begin{notation} \\label{T.N}\nLet  $t_1, \\dots , t_p$ denote the bits $j$ such that $y_j\\not=0$, assuming  $t_1 > \\dots > t_p$.\nSet $N = \\{1,2,\\ldots,n\\}$.\n\\end{notation}\n\n\nThe following algorithm defines index sets $\\emptyset=I_0\\subseteq I_1\\subseteq \\cdots I_p \\subseteq N$\nsuch that we can compute a move $x \\to x'$ with $M(x')=0$, by decreasing components $i\\in I_p$. Define $O_j=\\{i\\in I_j\\mid x_{it_{j+1}}=1\\}$, and set $\\ga_j = |I_j|$  and  $\\gb_j=|O_j|$.\n\n\n\n\\medskip\n\\noindent\n{\\bf Step 0}. Initialize $I_0=\\emptyset$\nand hence, $\\ga_0 = \\gb_0 = 0$, and set $x'_i:=x_i$ for all $i \\in N$.\n\n\\smallskip\n\\noindent\n{\\bf Step 1}. For $j=1, \\dots , p$, construct $I_j$ and update $x'$ as follows.\n\n\\smallskip\n\n{\\bf Case 1.} If $y_{t_j} \\leq \\beta_{j-1}$, then let  $I_j:=I_{j-1}$,\nchoose  $y_{t_j}$ many indices $i\\in O_{j-1}$,\nand update $x'_{it_j}:=0$.\n\n\\smallskip\n\n{\\bf Case 2.} If $y_{t_j} > \\beta_{j-1}$ and $(k+1) -y_{t_j} \\leq \\ga_{j-1}-\\gb_{j-1}$,  then\nlet  $I_j:=I_{j-1}$, choose  $(k+1)-y_{t_j}$ many indices $i\\in I_{j-1}\\setminus O_{j-1}$, and update $x'_{i{t_j}}:=1$.\n\n\\smallskip\n\n{\\bf Case 3.}  If $y_{t_j} > \\beta_{j-1}$ and $y_{t_j}-\\beta_{j-1} \\leq k-\\ga_{j-1}$, then\nlet $I_j$ be the index set obtained from $I_{j-1}$ by adding\n$y_{t_j} - \\beta_{j-1}$ many indices $i$ from $N \\setminus I_{j-1}$ such that\n$x_{i{t_j}}=1$.  Update $x'_{i{t_j}}:=0$ for $i\\in (I_j\\setminus I_{j-1})\\cup O_{j-1}$.\n\n\n\\medskip\n\\noindent\nWe first note that the three cases above are exclusive and\ncover all possible $y_{t_j}$, $\\ga_{j-1}$ and $\\gb_{j-1}$ values.\nMoreover, it is easily seen that\na position $x'$ after the execution of the algorithm satisfies  $M(x')=0$, and\n$x'_i=x_i$  holds for $i \\not\\in I_p$.\n\nNote that we increase the set $I_j$ only in Case 3, in which case we have\n\\[\n|I_j|=\\alpha_j=(y_{t_j}-\\beta_{j-1})+\\alpha_{j-1} \\leq k,\n\\]\nimplying $|I_p|\\leq k$.\n\nNote also that in Case 3 we must have at least $y_{t_j} - \\beta_{j-1}$ many indices $i\\in N\\setminus I_{j-1}$ with $x_{it_j}=1$ by the definition of $y_{t_j}$ in \\eqref{e0001}.\n\nIt remains to show that $x' < x$.\nAssume that $i$ is an index such that $x'_i \\not= x_i$.\nThen some $j$ satisfies $i \\not\\in I_{j-1}$ and $i \\in I_j$.\nThis implies that $x'_i$ was first updated during the $j$th iteration of Step 1.\nNamely, the $t_j$th bit of $x'_i$ is modified from $1$ to $0$.\nSince $x'_{it}=x_{it}$ holds for all $t$ with $t > t_j$,\nwe have $x'_i < x_i$, which completes the proof.\n\\qed\n\n\\subsection{Proof of Theorem \\ref{t-Moore-m=0,1} for $m=1$}\n\\label{ss-p-t1-1}\n\nNow, let us prove that\n$\\cG(x)=1$ if and only if  $M(x)=1$.\n\n\\smallskip\nThe proof in the previous subsection implies that for a position $x$ with $M(x)=1$ there exists a move $x\\to x'$ such that $M(x')=0$.\nBy the properties of the SG function, it remains to show that\n\\begin{itemize}\n\\item [\\rm{(i1)}] for any position $x$  with $M(x) = 1$, there exists\nno move $x \\to x'$ such that $M(x') = 1$;\n\\item [\\rm{(a1)}] for any position $x$ with $M(x) > 1$, there\nexists a move $x \\to x'$ such that  $M(x')=1$;\n\\end{itemize}\n\nWe prove \\rm{(i1)} similarly to (i0).\nLet us assume that $M(x)=1$ holds for a position $x$\nand consider a move $x \\to x'$.\nLet $j$ be the highest binary bit such that\n$x_{ij}$ and $x'_{ij}$ differ for some $i$.\nThen by Lemma \\ref{moore1} we have $x'_{ij} \\leq x_{ij}$ for all $i$,  and\n$1 \\leq \\sum_{i}(x_{ij}-x'_{ij}) \\leq k$.\nHence, $\\sum_{i}x'_{ij} \\not= \\sum_{i}x_{ij}  (\\mod (k+1))$ and, thus, $M(x') \\not= 1$.\n\n\nTo show \\rm{(a1)}, let us consider a position $x$  with $M(x) > 1$.\nSimilarly to  \\rm{(a0)}, we will algorithmically construct a move $x \\to x'$ such that $M(x')=1$.\n\nLet again $t_1, \\dots , t_{p-1}$ denote the bits $j>0$ such that $y_j>0$,\nwhere we assume that  $t_1 > \\dots > t_{p-1}$, and add $t_p=0$.\n\nThe algorithm remains the same, as for \\rm{(a0)},\nexcept for $j=p$, when $t_p=0$. We detail below the computation of $I_p$ from $I_{p-1}$:\n\n\n\\medskip\n\\noindent\n\n{\\bf Case 1.} If $y_0> 1$ and $y_0-1 \\leq \\gb_{p-1}$,\nthen let  $I_p:=I_{p-1}$,\nchoose  $y_0-1$ many indices ${i}$ from $I_p$  such that\n$x'_{i0}=1$, and update $x'_{i0}:=0$ for such indices.\n\n\\smallskip\n\n{\\bf Case 2.} If $y_0> 1$ and $(k+2)-y_0 \\leq \\ga_{p-1}-\\beta_{p-1}$,\nlet  $I_p:=I_{p-1}$, choose  $(k+2)-y_{0}$ many indices ${i}$ from $I_p$  such\nthat $x'_{i0}=0$ and update $x'_{i0}:= 1$ for such indices.\n\n\\smallskip\n\n{\\bf Case 3.} If $y_0> 1$ and $0< (y_0-1) -\\gb_{p-1} \\leq k-\\ga_{p-1}$, then\nlet $I_p$ be an index set obtained from $I_{p-1}$ by adding $(y_{0}-1)-\\beta_{p-1}$ many\nindices $i$ from $N \\setminus I_{p-1}$ such that $x_{i{0}}=1$, and\nupdate $x'_{i0}:=0$ for all $i \\in I_p$ with $x'_{i0}=1$.\n\n\\smallskip\n\n{\\bf Case 4.} If $y_0=0$ and $\\ga_{p-1} > \\gb_{p-1}$,\nthen let  $I_p:=I_{p-1}$,\nchoose  an index ${i}$ from $I_p$  such that $x'_{i0}=0$, and update $x'_{i0}:=1$.\n\n\\smallskip\n\n{\\bf Case 5.} If $y_0=0$ and $\\ga_{p-1} = \\gb_{p-1}=k$,\nthen let  $I_p:=I_{p-1}$, and update $x'_{i0}:=0$ for all $i \\in I_p$.\n\n\\smallskip\n\n{\\bf Case 6.} If $y_0=0$ and $\\ga_{p-1} = \\gb_{p-1}<k$,\nthen let $I_p$ be an index set obtained from $I_{p-1}$ by adding $k-\\ga_{p-1}$ many\nindices $i$ from $N \\setminus I_{p-1}$ such\nthat $x_{i{0}}=1$, and update $x'_{i0}:=0$ for all $i \\in I_p$.\n\n\\smallskip\n{\\bf Case 7.} If $y_0=1$, then we set $I_p=I_{p-1}$.\n\n\\medskip\n\\noindent\nNote that the above seven cases are exclusive and cover all possible $y_{0}$, $\\ga_{p-1}$, and $\\gb_{p-1}$ values.\nNote also that\nin Case 6,  $\\ga_{p-1} > 0$ since otherwise  $M(x)=0$, giving a contradiction.\nThus, in Case 6, we can choose $k - \\ga_{p-1}$ many\nindices $i$ from $N \\setminus I_{p-1}$ such that $x_{i{0}}=1$.\nLet $x'$ be a position obtained by the algorithm.\nThen, clearly  $M(x')=1$  and  $x \\to x'$ is a move. This completes the proof of (a1).\n\\qed\n\n\\section{More on the Tetris function} \\label{Ss.Tetris}\n\nIn this section we show that the SG function described in Theorems \\ref{thm.n<2k}\nand \\ref{thm.n=2k} can in fact be computed efficiently, in polynomial time. For this we need to show that the Tetris function can be computed in polynomial time for these games.\nWe also prove that for a given position $x$ and integer $0\\leq g < T_{n,k}(x)$\nwe can compute in polynomial time a move $x\\to x'$ such that $T_{n,k}(x')=g$.\nFinally, in subsection \\ref{tetris-degree} we recall\nsome relations to degree sequences of graphs and hypergraphs.\n\n\\subsection{Computing the Tetris function in polynomial time} \\label{comp-tet-in-p}\n\nLet us recall that to a position $x$ we associated a shifted position $\\bar x$ after Corollary \\ref{delta-epsilon} with the property that $T_{n,k}(x)=\\bar x_{n-k+1}$.\nThe procedure described there is a non-polynomial algorithm.\nHowever $\\bar x$ and consequently $T_{n,k}(x)=\\bar x_{n-k+1}$ can be computed in a more efficient way.\n\n\\begin{theorem}\\label{Tetrispolynomial}\nGiven a position $x$ we can compute $T_{n,k}(x)$ in linear time in $n$.\n\\end{theorem}\n\n\\proof\nWe can assume without loss of generality that $x$ is a nondecreasing position. We show that the corresponding $\\bar x$ can be constructed in linear time in $n$, and thus the claim follows by the equality $T_{n,k}(x)=\\bar x_{n-k+1}$.\n\nRecall that the input size is $\\log(\\prod_{i = 1}^n x_i)$. Let $s=\\sum_{i=1}^{n-k} x_i$ be the number of tokens we shift on top of the largest $k$ piles; see Figure \\ref{ADD(1)}.\nWe know that for some $\\ell <k$ the first $\\ell +1$ columns of $\\bar x$ have almost the same number of tokens (at most one difference.)\nTo determine this index $\\ell$ and the height of the resulting piles, we use simple volume based arguments.\nWe need to compute first the following parameters.\n\nFor each $i=1, \\ldots ,k-1$, we denote by $y_i=x_{n-k+i+1}-x_{n-k+i}$ the difference of the sizes of consecutive piles.\nSet $s_0 =0$, $s_{k}=\\infty$, and\nfor $i=1,...,k-1$, set $s_i=s_{i-1}+i \\cdot y_i$ (i.e., the number of tokens we need to shift on top of the first $i$ piles $(n-k+1), \\ldots, (n-k+i)$ to make them all equal to $x_{n-k+1+i}$.)\nWe define a unique $\\ell$ by $s_\\ell \\leq s < s_{\\ell+1}$.\nWe define $a = s - s_\\ell$, $\\alpha = \\lfloor\\frac{a}{\\ell+1}\\rfloor$, and $\\beta =a \\mod (\\ell+1)$.\nWe fill up the first $\\ell+1$ columns to level $x_{n-k+\\ell+1}$ using $s_\\ell$ tokens. Then,\nwe place the remaining $a$ tokens by increasing each of the first\n$\\ell+1$ columns (indexed $n-k+1,...,n-k+\\ell +1$) by $\\alpha$ and the last $\\beta$ of these by one more, as in the following expression.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[height=8cm]{Doc6}\n\\caption{An example of calculating $\\bar x$ for $x=(1,2,2,3,4,4,7)$ with $k = 4$.}\n\\end{figure}\n\n\n\\begin{align*}\n\\bar x_i=\n\\begin{cases}\n0,                        & \\text{if } i=1, \\ldots ,n-k; \\\\\nx_{n-k+\\ell+1}+\\alpha,    & \\text{if } i=n-k+1, \\ldots ,n-k+1+\\ell -\\beta; \\\\\nx_{n-k+\\ell+1}+\\alpha +1, & \\text{if } i=n-k+2+\\ell -\\beta, \\ldots, n-k+1+\\ell; \\\\\nx_i,                      & \\text{if } i=n-k+2+\\ell, \\ldots ,n.\n\\end{cases}\n\\end{align*}\nIt is easy to see that this defines $\\bar x$ correctly,\nand that all these parameters can be computed in $O(n)$ time, if $x$ is a nondecreasing vector.\n\\qed\n\n\\begin{remark}\nTechnically, only the computation of $s=\\sum_{i=1}^{n-k}x_i$ depends on $n$.\nAll other computations in the previous proof can be done in $O(k)$ time.\n\\end{remark}\n\\medskip\n\n\n\n\n\\subsection{Polynomial computation of a move to a given Tetris value} \\label{}\n\nLet $x = (x_1, \\ldots, x_n)$ be a nondecreasing position and $k > n\/2$.\nWe examine the question of how to move to some position of given Tetris value $g$.\n\nWe denote by\n\\begin{enumerate} \\itemsep0em\n\\item [\\rm{(i)}] $x^\\ell$ the position obtained from $x$ by removing all tokens from the largest $k$ piles of $x$, and by\n\\item [\\rm{(ii)}] $x^u$ the position obtained from $x$ by decreasing the largest $k$ piles by one unit each.\n\\end{enumerate}\n\nConsider the set\n$W = \\{T_{n,k}(x') \\mid \\forall x': x\\to x'\\}$.\nBy Lemma \\ref{tbasic} (i) we have $T_{n,k}(x^\\ell) = \\min(W)$.\nBy Lemma \\ref{bestslowmove} we have $T_{n,k}(x^u)=T_{n,k}(x)-1$. As in the proof of\nTheorem \\ref{thm.n<2k}, we can argue that for\nevery value $g$ such that $T_{n,k}(x^\\ell)\\leq g \\leq T_{n,k}(x)-1$ there exists a move $x\\to x'$ such that  $x^\\ell \\leq x' \\leq x$ and $T_{n,k}(x')=g$.\n\n\n\\begin{theorem}\\label{achieveTetrisvalue}  \nGiven $g \\in W$, computing a position $x'$ such that $x^\\ell \\leq x' \\leq x$ and $T_{n,k}(x')=g$ can be done in $O\\big(n\\log(\\sum_{i=n-k+1}^nx_i)\\big)$ time.\n\\end{theorem}\n\\proof\nWe have $T_{n,k}(x^\\ell)\\leq g \\leq T_{n,k}(x^u)$.\nUsing the monotonicity of the Tetris function we perform a binary search in the space of positions between $x^\\ell$ and $x^u$.\nIn a general step we compute $L=\\sum_{i=n-k+1}^nx^\\ell_i$ and $U=\\sum_{i=n-k+1}^nx^u_i$, set $M=\\lfloor\\frac{L+U}{2}\\rfloor$ and compute $y_i=int(\\frac{x^\\ell_i+x^u_i}{2})$ for $i=n-k+1$,\nwhere $int(\\cdot)$ is a rounding to a nearest integer value in such a way that $\\sum_{i=n-k+1}^ny_i=M$.\nFinally, we set $y_i=x_i$ for $i<n-k+1$.\nIf $T_{n,k}(y)<g$ then we replace $x^\\ell$ by $y$, otherwise we replace $x^u$ by $y$.\n\nClearly these computations can be done in each step in $O(n)$ time,\nand computing the Tetris value of $y$ can also be done in $O(n)$ time by Theorem~\\ref{Tetrispolynomial}.\n\\qed\n\n\\smallskip\n\n\\begin{remark}\nSimilarly to the proof of Proposition \\ref{Tetrispolynomial}\nwe could improve the complexity of the above algorithm to $O(n)$.\n\\end{remark}\n\n\\subsection{Tetris function and degree sequences of graphs and hypergraphs}\\label{tetris-degree}\n\nA related problem is the hypergraph realization of a given degree sequence.\nLet us fix $V=\\{1,2,...,n\\}$ as the set of vertices.\nA multi-hypergraph $\\cH=\\{H_1,...,H_m\\}$ is a family of subsets (called hyperedges) of $V$,\ni.e., $H_j\\subseteq V$ for all $j=1,...,m$. It is called $k$-uniform if $|H_j|=k$ for all $j=1,...,m$.\nThe degree $d_{\\cH}(i)$ of a vertex $i\\in V$ is the number of hyperedges $H_j$ of $\\cH$ that contain $i$.\nWe allow the same subset to appear multiple times in $\\cH$.\n\nGiven an integer vector $x\\in\\ZZ_{\\geq}^n$, one can ask if there exists a $k$-uniform multi-hypergraph $\\cH$ on the vertex set $V$ such that $d_{\\cH}(i)=x_i$ for all $i\\in V$.\nEquivalently, we examine the existence of a bipartite graph $G=(X,Y,E)$ such that $|X|=n$, $|Y|=m$, $d_G(i)=x_i$ for all $i\\in X$, and $d_G(j)=k$ for all $j\\in Y$.\nFor the latter we can apply the classical Gale-Ryser theorem claiming that the answer is yes if and only if\n\n\\begin{itemize} \\itemsep0em\n\\item [\\rm{(i)}]  $\\sum_{i=1}^nx_i=km$\n\\item [\\rm{(ii)}] $\\sum_{i=1}^n\\min(x_i,g) \\geq kg$ for all $g=1,...,m$.\n\\end{itemize}\nLet us note that checking these conditions may not be polynomial in $x$ and $k$, since $m=\\sum_{i=1}^nx_i\/k$ according to property \\rm{(i)}.\nLet us also note that following a sequence of slow moves starting from position $x$, each time the set of columns that we decrease by $1$ can be considered as a hyperedge of $\\cH$.\nThus a maximal sequence of slow moves will construct $\\cH$ if the Tetris function achieves its trivial bound $kT_{n,k}(x)=\\sum_{i=1}^nx_i$.\nThis equality is in fact equivalent with property \\rm{(i)}, since we must have $m=T_{n,k}(x) $ in this case.\nOur results in this section thus prove that for the above degree sequence realization problems the most efficient answer is to compute the Tetris function value in linear time, and then compare it to its trivial upper bound.\nIf these are the same then the answer is yes.\n\nHavel (1955) and Hakimi (1962) provided a simple greedy algorithm based on a characterization for the recognition of degree sequences of bipartite graphs.\nFor the above case their criterion states that $x$ is a degree sequence of a $k$-uniform multi-hypergraph if and only if the position $x'$ is also a degree sequence of a $k$-uniform multi-hypergraph, where $x'$ is obtained from $x$ by decreasing the $k$ largest components of $x$ by $1$.\nNote that this implies a recursive process that is one of the definitions we used for the Tetris function.\n\nLet us remark finally that in general $x$ is not the degree sequence of a $k$-uniform multi-hypergraph.\nIn this case however a move $x\\to x'$ such that $T_{n,k}(x')=0$ provides us with a minimal modification such that $x''=x-x'$ becomes the degree sequence of such a hypergraph.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe energy spectrum of primary cosmic rays\nextends over more than $12$ decades in energy.\nIt follows, over a large range, a simple power law\nindicating its non-thermal nature.\nThe highest energy cosmic rays (which means particles with energies above $10$~EeV $\\equiv 10^{19}$~eV)\nmust be produced by the most extreme non-thermal process in the Universe.\nTheir energy spectrum, arrival directions, and composition can be inferred from air shower observations.\nHowever, the origin of these particles is still unknown and there is not a consensus whether there is \na steepening in the energy spectrum at around $60$~EeV. \nDetermining the shape of the spectrum in this energy range in sufficient detail \nrequires an enormous detector exposure.\nThe Pierre Auger Observatory was designed to study these ultra-high energy cosmic rays.\nAlthough under construction, the Auger Southern site is the largest cosmic ray detector in the world.\n\nA fundamental characteristics of the Pierre Auger Observatory is its capability of \\textit{hybrid} reconstruction of cosmic ray \nshowers~\\cite{Mostafa:2005kd}. \nTwo independent detectors, the Surface Detector (SD), which samples the shower particles at ground, and the Fluorescence Detector (FD), \nwhich collects the fluorescence light emitted by the shower particles along their path in the atmosphere, \nare able to measure the energy and direction of the same cosmic ray shower.\nThe enhanced capabilities of the Auger hybrid detector are examined in this paper.\n\nThe Pierre Auger Observatory\nwas designed to observe, in coincidence, the shower particles at ground\nand the associated fluorescence light generated in the atmosphere.\nThis is\nachieved with a large array of water Cherenkov detectors coupled with\nair-fluorescence detectors that overlook the surface array.\nIt is not simply a dual experiment.\nApart from important cross-checks and measurement redundancy,\nthe two techniques see air showers in complementary ways.\nA single air shower is detected 3-dimensionally.\nThe ground array measures the 2-dimensional lateral structure of the shower at ground level,\nwith some ability to separate the electromagnetic and muon components.\nThe fluorescence detector records the longitudinal pro{f}ile of the shower during\nits development through the atmosphere.\n\nA \\textit{hybrid} event is an air shower that is simultaneously detected\nby the fluorescence detector and the ground array.\nThe Observatory was originally designed and is currently being built with a \\textit{cross--triggering}\ncapability.\nData are recovered from both detectors whenever either system is triggered.\nIf an air shower independently triggers both detectors the event is tagged\naccordingly.\nThere are also cases where the fluorescence detector,\nhaving a lower energy threshold,\npromotes a sub--threshold array trigger.\nSurface stations are then matched by timing and location.\nThis is an important capability because these sub--threshold hybrid events\nwould not have triggered the array otherwise.\nThe geometrical reconstruction of the air shower's axis is accomplished by minimizing a\n$\\chi^{2}$ function involving data from all triggered elements in the eye and on the ground.\nThe reconstruction accuracy is far better than the ground array counters and the single eye\ncould achieve independently~\\cite{Bonifazi:2005ns}.\n\nThe combination of the air fluorescence measurement and the particle detection on the ground provides\nan absolute energy calibration.\nThe FD measurements determine the longitudinal development profile,\nwhose integral is proportional to the total energy of the electromagnetic particle cascade.\nThe SD independently estimates the shower energy by evaluating the particle density at $1000$~m from the core.\nThis ground parameter, called $S(1000)$, is estimated from the surface stations by fitting the measured densities to a lateral\ndistribution function~\\cite{Ghia:2005bn}. \nA conversion factor that relates $S(1000)$ to the shower energy (based on the FD information) is extracted from hybrid events.\nThis reduces significantly the dependence on air shower models and on assumptions of the primary composition.\n\nThe Pierre Auger Collaboration started the construction of the Southern site in $2002$ in the city of \nMalarg\\\"{u}e, which is located at an elevation of $1400$~m a.s.l. in the province of Mendoza, Argentina.\nAfter a successful operation of a prototype experiment~\\cite{Abraham:2004dt},\nthe Southern Observatory started operation in \\textit{hybrid} production mode in January, 2004.\nSurface stations have a $100$\\% duty cycle,\nwhile fluorescence eyes can only operate on clear moonless nights.\nBoth surface and fluorescence detectors have been running simultaneously\n$14\\%$ of the time.\nThe number of hybrid events represents $10\\%$ the statistics of the surface array data.\nThe Southern site will be completed in 2007.\nAnother important objective is to obtain a uniform exposure over the full sky.\nThis will be achieved by constructing a second instrument in the Northern hemisphere.\nEach site is conceived to cover an area of $3000$~km$^{2}$ in order to collect about $1$ event\nper week and site above $60$~EeV.\nThe Northern Observatory is planned to be sited in the U.S.\n\n\n\n\\section{Hybrid Geometrical Reconstruction}\n\nA hybrid detector achieves the best geometrical accuracy by using timing information from all the detector components,\nboth FD pixels and SD counters.\nEach element records a pulse of light from which one can determine the central time of the pulse and its uncertainty.\nEach trial geometry for the shower axis yields a prediction for the times at each detector component.\nDifferences between actual and predicted times are weighted using their corresponding uncertainties,\nsquared, and summed to construct a $\\chi^{2}$ value.\nThe hypothesis with the minimum value of $\\chi^{2}$ is the reconstructed shower axis.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{golden_camera_bw.eps}\n\\caption{\nLight track of a hybrid event as seen by two adjacent fluorescence telescopes.\nThe full line is the fitted shower-detector plane.\nThe squares represent the surface detectors that also triggered in this event.%\n}\n\\label{fig:Gcamera} \n\\end{figure}\n\nIn the FD, cosmic ray showers are detected as a sequence of triggered pixels in the camera.\nAn example of an event propagating through two adjacent FD telescopes is presented in Fig.~\\ref{fig:Gcamera}.\nThe first step in the analysis is the determination of the shower-detector plane (SDP).\nThe SDP is the plane that includes the location of the eye and the line of the shower axis.\n(See the sketch in Fig.~\\ref{fig:sketch}.)\nExperimentally, it is the plane through the eye which most nearly contains the pointing directions of the FD pixels\ncentered on the shower axis.\n(See fitted line in Fig.~\\ref{fig:Gcamera}.)\nUsing a known axis provided from the Central Laser Facility (CLF), described in Ref.~\\cite{Arqueros:2005yn},\nthe SDP reconstruction error can be evaluated by comparing the space angle between\nthe normal vector to the experimentally determined SDP and the known true normal vector.\nThis uncertainty in the SDP is not greater than about $0.1^{\\circ}$.\n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{fd-geom_bw.eps}\n\\caption{Illustration of the geometrical shower reconstruction from the observables of the fluorescence detector.\n}\n\\label{fig:sketch} \n\\end{figure}\n\nNext, the timing information of the pixels is used for reconstructing the shower axis within the SDP.\nAs illustrated in Fig.~\\ref{fig:sketch}, the shower axis can be characterized by two parameters:\nthe perpendicular distance $R_{p}$ from the eye to the track,\nand the angle $\\psi$ that the track makes with the horizontal line in the SDP.\nEach pixel which observes the track has a pointing direction which makes an angle\n$\\chi_{i}$ with the horizontal line.\nLet $t_{0}$ be the time when the shower front on the axis passes the point of closest approach $R_{p}$ to the eye.\nThe light arrives at the $i$th pixel at the time \n\\begin{equation}\n\tt_i = t_0 + \\frac{R_p}{c} \\cot{[(\\psi+\\chi_i)\/2]}.\n\t\\label{eq:timefit}\n\\end{equation}\n\nThe shower parameters are then determined by fitting the data points to this functional form.\nUsing the FADC electronics, such a monocular reconstruction may achieve excellent accuracy.\nHowever, the accuracy of the monocular reconstruction is limited when the measured angular speed\n$d\\chi\/dt$ does not change much over the observed track length.\nAn example is shown in Fig.~\\ref{fig:timefit}.\nFor these events (usually short tracks) there is a small curvature in the functional form of Eq.~(\\ref{eq:timefit})\nsuch that there is a family of possible $(R_p,\\psi)$ axis solutions.\nThis is also illustrated in Fig.~\\ref{fig:correlation}.\nNot only the uncertainties in $R_p$ and $\\psi$ are large,\nbut also they are strongly correlated.\nThis translates directly into an uncertainty in the other shower parameters,\nespecially in the reconstructed shower energy.\nThis asymmetric uncertainty in the energy and angular resolution are important drawbacks of the\nmonocular reconstruction.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{timefit_bw.eps}\n\\caption{Functional form that correlates the time of arrival of the light at each pixel with the \nangle between the pointing direction of that particular pixel and the horizontal line within the shower-detector plane.\n}\n\\label{fig:timefit} \n\\end{figure}\n\nThe aforementioned degeneracy can be broken by combining\nthe timing information from the SD stations with that of the FD telescopes.\nThis is called the \\textit{hybrid} reconstruction.\nThe hybrid solution \nfor the example shown in Fig.~\\ref{fig:timefit} \nis shown in Fig.~\\ref{fig:correlation}\nas a white star and its uncertainty as the smaller (full) ellipse.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{correlation_bw.eps}\n\\caption{Solutions (and $1\\sigma$ regions) for the axis fit shown in Fig.~\\ref{fig:timefit}.\nThe large uncertainty (and strong correlation) of the monocular reconstruction is broken using the \ntiming information from the surface detectors (Hybrid).\nThe stars mark the solutions that minimize the $\\chi^2$ for the axis reconstruction.\n(Following the notation of this paper,\n$\\chi_0 = 180^{\\circ} - \\psi$.)\n}\n\\label{fig:correlation} \n\\end{figure}\n\nSince the SD operates with a $100\\%$ duty cycle,\nmost of the events observed by the FD are in fact hybrid events.\nThere are also cases where the fluorescence detector,\nhaving a lower energy threshold,\npromotes a sub--threshold array trigger.\nSurface stations are then matched by timing and location.\nThis is an important capability because these sub--threshold hybrid events\nwould not have triggered the array otherwise.\nIn fact, the time of arrival at a single counter at ground can suffice for the\nhybrid reconstruction. \n\nThe reconstruction uncertainties are evaluated\nusing events with \\textit{known} geometries, \\textit{i.e.} laser beams.\nSince the location of the CLF \n(approximately equidistant from the first three fluorescence sites)\nand the direction of the laser beam are\nknown to an accuracy better than the expected angular resolution of the\nfluorescence detector, laser shots from the CLF can be used to measure the accuracy of\nthe geometrical reconstruction.\nFurthermore, the laser beam is split and part of the laser light is sent\nthrough an optical fiber to a nearby ground array station.\nThus, the axis of the laser light can be reconstructed both in monocular mode and in the\n\\textit{single-tank} hybrid mode.\nThe resolution of the monocular and hybrid reconstructions are compared in\nFig.~\\ref{RESOLVRP} for the distance between the eye and the CLF, and\nin Fig.~\\ref{RESOLVCHI0} for the angle of the axis.\nThe results are very encouraging.\nWith the monocular reconstruction, the location of the CLF can be determined\nwith a resolution of $\\sim500$~m.\nAfter including the timing information of the single water tank,\nthe resolution improves by one order of magnitude with no systematic shift.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{resolRp_bw.eps}\n\\caption{\\label{RESOLVRP} Difference between the reconstructed and true distance\nfrom the eye to the vertical laser beam using the monocular and hybrid techniques.\nThe location of the laser is known to $5$~m.\n}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{resolChi0_bw.eps}\n\\caption{\\label{RESOLVCHI0} Angular difference between reconstructed and true direction\nof the laser beam using the monocular and hybrid techniques.\nThe laser beam is vertical within $0.01^{\\circ}$.\n}\n\\end{figure}\n\nAs mentioned before, the laser light from the CLF produces simultaneous triggers in both the surface and\n(three) fluorescence detectors. \nThe recorded event times are used to measure and monitor the\nrelative timing between the two detectors.\nThe time offset between the first fluorescence eye and the surface detector is shown in\nFig.~\\ref{OFFSET}.\nThis time offset is crucial for the accuracy of the hybrid reconstruction, \nand it has been measured to\nbetter than $50$~ns~\\cite{Allison:2005vj}.\nThe contribution to the systematic uncertainty in the core location due\nto the uncertainty in the time synchronization is $20$~m.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{offset.eps}\n\\caption{\\label{OFFSET} Time offset between the surface detector and one of the fluorescence detectors.\nThe variation in time is within the uncertainty of $50$~ns.\n}\n\\end{figure}\n\nUsing the timing information from the eye's pixels together with the surface stations\nto reconstruct real air showers,\na core location resolution of $50$~m is achieved.\nThe resolution for the arrival direction of cosmic rays is\n$0.6^{\\circ}$~\\cite{Bonifazi:2005ns}.\nThese results for the \\textit{hybrid} accuracy are in good agreement with\nestimations using analytic arguments~\\cite{Sommers:1995dm},\nmeasurements on real data using a bootstrap method~\\cite{Fick:2003qp},\nand previous simulation studies~\\cite{Dawson:1996ci}.\n\n\\section{Energy Determination}\n\nAfter successful reconstruction of the event geometry,\nthe FADC traces of the FD pixels are analyzed in order to obtain the light emitted along the shower axis.\nAn atmospheric scattering model is used in this step to transform the light received at the FD back to the light emitted\nfrom the shower axis.\nThe geometrical height, as observed by the telescopes, is converted to grammage of atmosphere.\nThe amount of fluorescence light emitted from a volume of air is proportional to the energy dissipated by the shower\nparticles in that volume.\nThe observed longitudinal light profile represents the energy loss in the atmosphere, which in turn is proportional\nto the number of charged particles in a given volume.\nThe result for a hybrid shower of $\\theta\\sim60^{\\circ}$ \nis shown in Fig.~\\ref{fig:profile} (bottom).\nThe line represents the best fit to a Gaisser-Hillas function~\\cite{GH},\nyielding a primary energy of $E=(23\\pm6)$~EeV, in good agreement with the $S(1000)$ determination. \n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.45\\textwidth]{profile_bw.eps}\n\\caption{\\label{fig:profile} An air-shower event detected both in water tanks (\\textit{top:} array view), \nand in a fluorescence detector (\\textit{bottom:} longitudinal profile).\n}\n\\end{figure}\n\n\n\n\\section{Observatory Status}\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[width=0.45\\textwidth]{status_bw.eps}}\n\\caption{\\label{STATUS} Status of the Pierre Auger Southern Observatory on May 10, 2006.\nThere were 1138 deployed tanks\nand 953 of them were fully operational.\nAlso shown are the locations of \nthe fluorescence detectors with their fields of view \n(for the three fully operational detectors).\nThe fourth fluorescence site (Loma Amarilla) is under construction.\n}\n\\end{figure}\n\nThe status of the Southern Observatory on May 10, 2006 is summarized in Fig.~\\ref{STATUS}.\nThere were 953 fully operational tanks at that time.\nThe {f}irst two fluorescence sites (Los Leones and Coihueco) were fully operational,\n\\textit{i.e.} running six telescopes each, in June, 2004.\nThe third site (Los Morados) started operation on March 18, 2005.\nThe fourth and last site (Loma Amarilla) is currently under construction and is scheduled\nto start operation by the end of 2006.\nThe present average rate is $50$ hybrid events per night per eye,\nfor a total of $\\sim40000$ events up to June, 2006.\nAt this rate, $4000$ hybrid events per month are expected\nwhen the Southern site is completed.\n\n\\section{Hybrid Measurements}\n\nIn the Pierre Auger Observatory two complementary experimental approaches\nare combined on a shower-by-shower basis within one single experiment.\nSuch redundancy allows cross-correlations between experimental techniques,\nthereby controlling the systematic uncertainties.\nFurthermore, there is an improvement in the resolution of the energy, mass,\nand arrival direction of the reconstructed primary particles.\nData are being used to verify both the performance of the individual components~\\cite{Bellido:2005aw,Bertou:2005bx},\nas well as to produce the highest quality reconstructed air-shower events~\\cite{Matthews:2005ve}.\nResults are very promising~\\cite{Mantsch:2005uq}\nand underline the advantages of the hybrid approach.\n\n\\subsection{Energy Spectrum}\n\nHybrid air shower measurements \nare utilized in the spectrum analysis to avoid dependence on\nspecific numerical simulations of air showers and detector\nresponses to them.  The analysis is also free of assumptions\nabout the primary nuclear masses.  The FD\nprovides a nearly calorimetric, model-independent energy\nmeasurement: fluorescence light is produced in proportion to\nenergy dissipation by a shower in the atmosphere.\nHybrid data establish the relation of shower energy to a ground parameter. \nMoreover,\nhybrid data determine the trigger probability for individual\ntanks as a function of core distance and energy, from which it\nis found that the SD event trigger is fully efficient above $3$~EeV\nfor zenith angles less than $60^{\\circ}$.  The SD exposure is\nthen calculated simply by integrating the geometric aperture over\ntime.\nIt is the continuously operating surface array which provides the high\nstatistics with unambiguous exposure.\nAssigning energies to the SD event set is a two-step process.\nThe first step is to assign an energy parameter, $S_{38}$, to each\nevent.  Then the hybrid events are used to establish\nthe rule for converting $S_{38}$ to energy.\n\nThe energy parameter $S_{38}$ for each shower comes from its\nexperimentally measured $S(1000)$.\nIt \nmay be regarded as the $S(1000)$\nmeasurement the shower would have produced if it had arrived\n$38^{\\circ}$ from the zenith.\nAs it can be seen in Fig.~\\ref{fig:cic}, \n$S_{38}$ is well correlated with the FD energy measurements of \nhigh quality hybrid events.\nThe fitted line gives an empirical rule for\nassigning energies (in EeV) based on $S_{38}$ (in VEM):\n\\begin{equation}E = 0.16 \\times S_{38}^{1.06}.\\end{equation}\nThe SD acceptance is not saturated below $3$~EeV. \nBut the hybrid events used in Fig.~\\ref{fig:cic},  \nwhich start at $\\sim1$~EeV,  \nare those\nwith core locations and arrival directions such that they have\na probability greater than 0.9 for satisfying the SD trigger and\nquality conditions.  \nThe distribution over energy produced by this two-step procedure\nbecomes the cosmic ray spectrum presented in Ref.~\\cite{Sommers:2005vs}.\n\n\\begin{figure}[!ht]\n\\includegraphics*[width=0.45\\textwidth]{calibrate.eps}\n\\caption{\\label{fig:cic} Shower energy (measured from the longitudinal profile) as a function of the ground parameter\n  $S_{38}$.  Each point is a hybrid event recorded when\n  there were contemporaneous aerosol measurements, whose \n  longitudinal profile included shower maximum in a measured\n  range of at least 350~g\\ cm$^{-2}$, and in which there was less than 10\\%\n  Cherenkov contamination.  The fitted line is\n  $\\log(E)=-0.79+1.06\\ \\log(S_{38})$.}\n\\end{figure}\n\n\\subsection{Anisotropy}\n\nDue to the much improved angular accuracy,\nthe \\textit{hybrid} data sample is ideal for anisotropy studies\nand, in particular, for point source searches.\nResults on a search for a point--like source in the direction of the Galactic\nCenter (GC) using these hybrid events were presented in Ref.~\\cite{Aglietta:2006ur}.\nThese events have a better angular resolution\n($ 0.7^\\circ$ at 68\\% c.l. in the EeV energy range).\nConsidering the hybrid events with an energy between $10^{17.9}$~eV and $10^{18.5}$~eV,\nno significant excess is seen in the GC direction.\nFor instance, in an optimal top-hat window of\n$1.59\\sigma\\simeq 0.75^\\circ$ radius,\n 0.3 events are expected while\nno single event direction falls within that circle.\nThis leads to a source flux upper-bound at 95\\% c.l. of\n\\begin{equation}\n\\Phi_s^{95}=\\xi\\; 0.15 \\ {\\rm km^{-2}\\ yr^{-1}}.\n\\end{equation}\n(The factor $\\xi$ is close to unity and parametrises the\nuncertainties in the flux normalization.)\n\nNote that\nthe energy assignments of the FD apply regardless of the assumed primary\ncomposition (except for a small correction to account for the missing energy),\nbe they protons or heavy nuclei. However,\nthe acceptance has  a dependence on composition because different primaries\ndevelop at different depths in the atmosphere. Since a quality\nrequirement for hybrid events is to have the maximum of the shower\ndevelopment inside the field of view of the telescopes, this affects\nthe sensitivity to different primaries.\nThe bound obtained is indeed conservative even if the bulk of the cosmic rays are\nheavy nuclei.\n\n\\subsection{Composition}\n\nAn upper limit for the photon fraction in cosmic rays with energies above $10^{19}$~eV\nwas derived using the hybrid data set~\\cite{Abraham:2006ar}.\nThe importance of the photon fraction lies in the fact that \\textit{top-down}\nmodels predict a considerable proportion of photons among the generated particles.\nThe measured photon flux is thus a valuable indicator of these non-accelerator models.\n\nThe method used in Ref.~\\cite{Abraham:2006ar} to distinguish between hadrons and photons\nin the hybrid data exploits the information on the longitudinal profile of the air shower.\nIn fact, this is the first such limit on photons obtained by observing the fluorescence light\nprofile of air showers.\nThe atmospheric depth at the shower maximum, $X_{max}$, is commonly used as a discriminant observable\nfor the cosmic ray composition because\nlighter nuclei penetrate, on average, more deeply into the atmosphere.\nAbove $10^{19}$~eV, showers initiated by photons develop significantly deeper in the atmosphere than hadronic\nshowers.\n\nNo candidate for a primary photon was found in the hybrid data taken between Jan, 2004 and Feb, 2006.\nBy comparing the observed $X_{max}$ of each hybrid event to predictions from hadronic simulations,\nan upper limit of $16\\%$ (at $95\\%$ c.l.) was derived.\nThis results confirms and improves the existing limits above $10$~EeV.\n\nThis analysis is currently limited mainly by the the small number of events.\nThe number of hybrid events will considerably increase in the next few years,\nand much lower primary photon fractions can be tested.\nAn upper limit of $\\sim5\\%$ could be achieved, for example, with two more years of data taking.\nMoreover, the larger statistics will allow to increase the threshold energy\nabove $10$~EeV where even larger photon fractions are predicted by the models.\nA similar limit ($\\sim15\\%$) but at higher energy (e.g. above $40$~EeV), would be well below existing limits\nand severely constrain non-acceleration models.\n\n\\section{Conclusions}\n\nThe Pierre Auger Observatory is a hybrid detector with excellent capabilities\nfor studying the highest energy cosmic rays.\nMuch of its capability stems from the accurate geometric reconstruction it achieves.\nThe shower geometry is reconstructed combining information from the eyes and the ground detectors.\nArrival directions are determined to a small fraction of a degree and the shower core is located to an\naccuracy of about $60$~m~\\cite{Bellido:2005aw,Bonifazi:2005ns}.\n\nThe construction of the Southern Observatory is well under way.\nEighteen FD telescopes and more than $60\\%$ of the surface array are in operation\ntaking data routinely.\nAt the present rate of deployment,\nconstruction will be finish in mid 2007.\nDetectors are performing very well and the first results are very encouraging.\n\nEmphasis is placed on hybrid analysis that provide unprecedented quality\nin geometry, energy, and mass reconstruction.\nOf utmost importance for the near future will be the determination of the energy spectrum\nto study the GZK feature, and the search for anisotropies in arrival directions.\nIt is important to note that\nboth SD and FD techniques have different systematics,\nand results are preliminary at this stage while the Observatory is under construction.\nThe possibility of studying\nthe same set of air showers with two independent methods is\nvaluable in understanding the strengths and limitations of each technique.\nThe hybrid analysis bene{f}its from the calorimetry of the fluorescence technique\nand the uniformity of the surface detector aperture.\n\nIn parallel to the completion of the Southern Observatory and to the analysis of data towards\nthe first scientific results, R\\&D has started for the development of the Northern site in the US.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sintro}\n\nSince the nineteenth century, $j$-invariants associated with elliptic curves having complex multiplication (CM), the so-called singular moduli, have been an object of study in number theory. A theorem of Weber \\cite[Theorem~11.1]{Cox} states that every singular modulus is an algebraic integer. Under certain technical restrictions, Gross and Zagier \\cite{Gross1985} stated explicit formulas for the absolute norm of the difference between two singular moduli.\n\nMotivated by effective results of Andr\\'e-Oort type \\cite{BMZ13,\n  Ku12}, David Masser raised in 2011 the question whether only\nfinitely many singular moduli are algebraic units, that is, units of the\nring of all algebraic integers. Throughout this article, we call such\nhypothetical algebraic numbers \\textsl{singular units}. Since there\nis no example of a singular unit in the literature, it seems\nlegitimate to ask whether there are any singular units at all.\n\nIn~\\cite{hab:junit}, the second-named author answered Masser's original  question in the affirmative: There exist at most finitely many singular units. However, his proof is non-effective as it invokes Siegel's lower bounds on the class number of imaginary quadratic fields \\cite{Siegel1935} through Duke's Equidistribution Theorem \\cite{Duke:hyperbolic}.\n\nHere, we can give the following definite answer to Masser's question as our main theorem.\n\n\\begin{theorem}\n\\label{thmain}\nThere are no singular units.\n\\end{theorem}\n\n\n\n\nTheorem~\\ref{thmain} is a formal consequence of our Theorems~\\ref{thhighrange},~\\ref{thmidrange},~\\ref{thlowrange} and~\\ref{thm:habeggerspari}. Let us briefly \nsketch its proof.\nWe say that a\nsingular modulus is of discriminant~$\\Delta$ if it is the\n$j$-invariant of a CM-elliptic curve whose endomorphism ring is the\nimaginary quadratic order of discriminant~$\\Delta$. We also write\n${\\Delta=Df^2}$ where~$D$ is the discriminant of the CM-field\n$\\Q(\\sqrt\\Delta)$, the fundamental discriminant, and $f$ is the\nconductor of the endomorphism order. The singular moduli of a given\ndiscriminant $\\Delta$ form a full Galois orbit over~$\\Q$ of\ncardinality equal to the class number\\footnote{We do not use the more traditional notation $h(\\Delta)$ because of the risk of confusing it with the height ${\\mathrm{h}}(\\cdot)$.} ${\\mathcal{C}}(\\Delta)$. \n\nWrite $\\zeta_3$ (resp.\\ $\\zeta_6$) for the third (resp.\\ sixth) root\nof unity $e^{2\\pi i\/3}$ (resp.\\ $e^{\\pi i\/3}$). Note that $\\zeta_3$\n(resp.\\ $\\zeta_6$) is the left (resp.\\ right) vertex of the geodesic\ntriangle enclosing the standard fundamental domain $\\mathcal{F}$ in\nthe Poincar\u00e9 upper half-plane. Given ${\\varepsilon\\in (0,1\/3]}$, denote by\n  ${\\mathcal{C}}_\\varepsilon(\\Delta)$ the number of singular moduli of\n  discriminant~$\\Delta$ which can be written ${j(\\tau)}$ where~$\\tau\n  \\in \\mathcal{F}$ satisfies ${|\\tau-\\zeta_3|< \\varepsilon}$ or\n  ${|\\tau-\\zeta_6|< \\varepsilon}$ and ${j(\\cdot)}$ denotes Klein's $j$-function. Since $\\zeta_3$ and $\\zeta_6$ are the only\n  zeros  of the $j$-function contained in the closure of\n  $\\mathcal{F}$, a pivotal ingredient in the proof of Theorem\n  \\ref{thmain} is an upper bound on ${\\mathcal{C}}_\\varepsilon(\\Delta)$. Indeed, a main\n  point of the argument in \\cite{hab:junit} is the\n  estimate\n  $${{\\mathcal{C}}_\\varepsilon(\\Delta)\\ll {\\mathcal{C}}(\\Delta)\\varepsilon^2}$$\nwhich holds when~$|\\Delta|$ is sufficiently large (in terms of~$\\varepsilon$). Unfortunately, ``sufficiently large'' here\n   is not effective; in fact, this is the place  where \n Duke's Equidistribution Theorem~\\cite{Duke:hyperbolic}, generalized by Clozel and Ullmo~\\cite{CU:Equidistribution} to arbitrary discriminants, is used.\n\nOur main novelty is the following effective estimate (see Theorem~\\ref{thceps}):\n\\begin{equation}\n\\label{ecepsrough}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\ll F\\left(\\frac{\\sigma_1(f)}{f}|\\Delta|^{1\/2}\\varepsilon^2+|\\Delta|^{1\/2}\\varepsilon+ \\sigma_0(f)  |\n\\Delta|^{1\/4}\\varepsilon+1\\right),\n\\end{equation}\nwhere\n$$\nF=F(\\Delta) = \\max \\bigl\\{2^{\\omega(a)}: a\\le {|\\Delta|^{1\/2}}\\bigr\\}.\n$$\nHere  and in the sequel all implicit constants are effective,  and we use the standard notation\n\\begin{align}\n  \\label{eq:standardnotation}\n\\omega(n)=\\sum_{p\\mid n}1,\\quad \\sigma_0(n)=\\sum_{d\\mid n}1,\\quad \\sigma_1(n)=\\sum_{d\\mid n}d. \n\\end{align}\nUsing that\n\\begin{equation}\n\\label{eolog}\n\\omega(n)=o(\\log n),\\quad \\log\\sigma_0(n)=o(\\log n), \\quad \\sigma_1(n)\\ll n\\log\\log n,\n\\end{equation}\nwe deduce from~\\eqref{ecepsrough} tha\n\\begin{equation}\n\\label{ecepseta}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\le |\\Delta|^{1\/2+o(1)}\\varepsilon+  |\n\\Delta|^{o(1)}\n\\end{equation}\nas ${|\\Delta|\\to\\infty}$. \n\nThe height of a singular unit~$\\alpha$ of discriminant~$\\Delta$ can be easily estimated in terms of ${\\mathcal{C}}_\\varepsilon(\\Delta)$:\n\\begin{equation}\n\\label{eupperheight}\n{\\mathrm{h}}(\\alpha)\\ll\\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta)}\\log|\\Delta|+\\log(\\varepsilon^{-1}),\n\\end{equation}\nsee Theorem~\\ref{theceps}.\n(By the \\textit{height} we mean here the usual \\textit{absolute logarithmic height} of an algebraic number; its definition is recalled in the beginning of Section~\\ref{supper}.) Substituting \\eqref{ecepseta}, we obtain the following upper estimate:\n\\begin{equation*}\n{\\mathrm{h}}(\\alpha)\\le \\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}A\\varepsilon+  \\frac{|\n\\Delta|^{o(1)}}{{\\mathcal{C}}(\\Delta)}+O(\\log(\\varepsilon^{-1}))\n\\end{equation*}\nwhere ${A=A(\\Delta)=|\\Delta|^{o(1)}}$ \nas ${|\\Delta|\\to\\infty}$. Specifying \n$$\n\\varepsilon= \\frac{{\\mathcal{C}}(\\Delta)}{A|\\Delta|^{1\/2}}\n$$\n(which is a nearly optimal choice), we obtain the estimate \n$$\n{\\mathrm{h}}(\\alpha)\\le   \\frac{|\n\\Delta|^{o(1)}}{{\\mathcal{C}}(\\Delta)}+o(\\log|\\Delta|) +O\\left(\\log^+\\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}\\right)\n$$\nwhere ${\\log^+(x)=\\max\\{\\log x,0\\}}$. \n\nTo obtain an upper bound on $|\\Delta|$, we combine this bound with the following two lower estimates on ${\\mathrm{h}}(\\alpha)$ (see Section \\ref{slower})\n\\begin{align}\n\\label{ecolm}\n{\\mathrm{h}}(\\alpha)+1&\\gg \\log|\\Delta|, \\\\\n\\label{etriv}\n{\\mathrm{h}}(\\alpha)&\\gg \\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}.\n\\end{align}\nThe bound~\\eqref{ecolm} is rather deep and relies on work of Colmez~\\cite{Colmez} and Nakkajima-Taguchi \\cite{NT}. On\nthe contrary,~\\eqref{etriv}  follows easily from the fact that one of\nthe conjugates of our singular unit~$\\alpha$ is\n${j\\bigl((\\Delta+\\sqrt\\Delta)\/2\\bigr)}$. Nevertheless, \\eqref{etriv} plays a\ncrucial role when the class number is pathologically small so that it\nwould contradict the Generalized Riemann Hypothesis (GRH). In fact, \\eqref{etriv} becomes much stronger than \\eqref{ecolm} in these hypothetical cases.\n\nComparing upper and lower estimates, we obtain for large $|\\Delta|$ that\n$$\n\\max\\left\\{\\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}, \\log|\\Delta|\\right\\}\\le \\frac{|\n\\Delta|^{o(1)}}{{\\mathcal{C}}(\\Delta)}+o(\\log|\\Delta|) +O\\left(\\log^+\\frac{|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}\\right),\n$$\nwhich is clearly impossible.\n\nTo get an explicit bound on $|\\Delta|$, we need to replace all implicit constants above with explicit ones. This relies in particular on a numerically sharp estimate for the arithmetic function $\\omega(n)$ due to Robin~\\cite{Ro83}. In Section \\ref{stenfourteen}, we see that this leads to a bound $|\\Delta|< 10^{15}$. While already  effective, it is still not feasible to check directly by a computer-assisted proof that none of the singular moduli of discriminant ${\\Delta \\in (- 10^{15}, -3]}$ is an algebraic unit.\n\nA refinement of our original arguments comes to our rescue. When  ${|\\Delta|<10^{15}}$, we improve on the estimate~(\\ref{ecepsrough}) by bounding sums of the form $\\sum_{n \\in [a,b] \\cap \\Z} 2^{\\omega(n)}$ in a more refined way. A natural idea is to use the Selberg-Delange method, which yields the asymptotic expansion\n\\begin{equation}\n\\label{equation::selbergdelange}\nS(x)= \\sum_{n\\le x} 2^{\\omega(n)} = x \\log x\\left(\\lambda_0 + \\frac{\\lambda_1}{\\log x} + O\\bigl(e^{-c\\sqrt{\\log x}}\\bigr)\\right)\n\\end{equation} \nwith explicit constants $\\lambda_0,\\lambda_1 \\in \\mathbb{R}$ and some\n constant ${c>0}$ (see, for instance,  \\cite[Theorem\n  II.6.1]{Te15}). There are two downsides of this method. First, the\nerror term is suboptimal under assumption of the GRH. Second, it would need some effort to make the constant $c$ actually explicit. \n\nHowever, as $|\\Delta|<10^{15}$, we are only interested in the case\nwhere $[a,b]$ is a subinterval of  ${[1, 2 \\cdot 10^7]}$. In this\nrange, a simple \\textsf{SAGE} script using the \\textsf{MPFI} library\n\\cite{MPFI,sagemath} can be used to improve on\n\\eqref{equation::selbergdelange}  computationally (see Proposition\n\\ref{proposition::selbergdelange}). As a consequence, we obtain\n$|\\Delta|<10^{10}$ for any singular unit of discriminant $\\Delta$ in Theorem \\ref{thmidrange}.\n\nThis is still not sufficient to check all remaining cases, at least\nwith modest computational means. The range is nevertheless small\nenough to use a counting algorithm in order to bound\n${\\mathcal{C}}_{10^{-3}}(\\Delta)$ for all discriminants $\\Delta$ satisfying\n$|\\Delta|< 10^{10}$, see Lemma \\ref{lem:lowrangeCebound}. {This still\nneeds an appropriate counting strategy, as determining\n${\\mathcal{C}}_{10^{-3}}(\\Delta)$ for each discriminant is rather slow,\ncomparable to computing separately each class number ${\\mathcal{C}}(\\Delta)$ in the\nsame range. Our trick is to bound all ${\\mathcal{C}}_{10^{-3}}(\\Delta)$\nsimultaneously by running through a set containing all imaginary quadratic\n$\\tau \\in \\mathcal{F}$ satisfying ${|\\tau-\\zeta_3|< \\varepsilon}$ or\n${|\\tau-\\zeta_6|< \\varepsilon}$ and such that $j(\\tau)$ is of discriminant\n$\\Delta$ with $|\\Delta|<10^{10}$. For each $\\tau$ encountered, we\ncompute its discriminant $\\Delta(\\tau)$ after the fact and increment\nour counter for ${\\mathcal{C}}_{10^{-3}}(\\Delta(\\tau))$. The thus obtained\nbounds for ${\\mathcal{C}}_{10^{-3}}(\\Delta)$ refine once again our previous\ninequalities, and allow us to conclude that $|\\Delta|<10^7$. Repeating this\nprocedure once again, with a slightly changed $\\varepsilon$, we\nachieve even $|\\Delta|<3 \\cdot 10^5$ in Theorem \\ref{thlowrange}.\nThese remaining cases can now be dealt with directly, for which we use\na \\textsf{PARI}~\\cite{PARI} program to prove Theorem  \\ref{thm:habeggerspari}, completing thereby the proof of Theorem~\\ref{thmain}. \n\n\\bigskip\n\nIt is very probable that our argument can be adapted to solve a more general problem: given an algebraic integer~$\\beta$,  determine the singular moduli~$\\alpha$ such that ${\\alpha-\\beta}$ is a unit; or at least bound effectively the discriminants of such~$\\alpha$. For instance, one may ask whether~$0$ is the only singular modulus~$\\alpha$ such that ${\\alpha-1}$ is a unit. In the general case, as explained in~\\cite{hab:junit}, this would require lower bounds for elliptic logarithmic forms, but when~$\\beta$ itself is a singular modulus, our argument extends almost without changes. One may go further and obtain an effective version of Theorem~2 from~\\cite{hab:junit}, which is an analogue of Siegel's Finiteness  Theorem for\nspecial points.\n\n\nThe famous work of Gross-Zagier and Dorman \\cite{Dorman1988,Gross1985} inspires the following problem: determine all couples $(\\alpha,\\beta)$ of singular moduli such that ${\\alpha-\\beta}$ is a unit; presumably, there is none. As indicated above, when~$\\beta$ is fixed and~$\\alpha$ varying, a version of our argument does the job, but if we let both~$\\alpha$ and~$\\beta$ vary, the problem seems more intricate. Very recently Yingkun Li~\\cite{Li18} made important progress: he proved that ${\\alpha-\\beta}$ is not a unit if the discriminants of~$\\alpha$ and~$\\beta$ are  fundamental and coprime. In particular, his result implies the following partial version of our Theorem~\\ref{thmain}: the discriminant of a singular unit must be either non-fundamental or divisible by~$3$. \n\nAnother natural problem is extending our work to $S$-units. Recall that, given a finite set~$S$ of prime numbers, a non-zero algebraic number is called an $S$-unit if both its denominator and numerator are composed of prime ideals dividing primes from~$S$.  Recently\nHerrero,\nMenares and\nRivera-Letelier announced the proof of finiteness of the set of singular $S$-units (that is, singular moduli that are $S$-units) for any finite set of primes~$S$. However, to the best of our knowledge, their argument is not effective as of now. \n\n\n\\bigskip\n\nFinally, let us discuss an application of Theorem~\\ref{thmain} to\neffective results of Andr\\'e-Oort type. A point\n$(\\alpha_1,\\ldots,\\alpha_n) \\in \\mathbb{C}^n$ is called special if\neach $\\alpha_i$, $i \\in \\{ 1, \\dots, n\\}$, is a  singular modulus. Since singular moduli are algebraic integers, the following statement is an immediate consequence of our main result. \n\n\n\n\n\\begin{corollary} For each polynomial ${P}$ in unknowns $X_2,\\ldots,\n  X_n$ and  coefficients that are algebraic integers in~$\\C$,\n \n  the hypersurface defined by \n\\begin{equation*}\nX_1 P(X_1, \\dots, X_n) = 1\n\\end{equation*}\n contains no special points.\n\\end{corollary}\n\nIn particular, $\\alpha_1^{a_1}\\cdots \\alpha_n^{a_n}\\not=1$\nfor all special points $(\\alpha_1,\\ldots,\\alpha_n)$\nand all integers $a_1\\ge 1,\\ldots,a_n \\ge 1$. \nThis corollary exhibits a rather general class of algebraic varieties\nof arbitrary dimension and degree for which  the celebrated theorem of\nPila~\\cite{Pila2011a} can be proved effectively and even explicitly.\nIt is complementary to other recent effective results of Andr\\'e-Oort type~\\cite{Bilu2017,Bi18}.\n\n\\paragraph{Plan of the article}\nIn Section~\\ref{sceps} we obtain an explicit version of the\nestimate~\\eqref{ecepsrough}. In Section~\\ref{supper} we obtain an\nupper estimate  for the height of a singular unit. In\nSection~\\ref{slower} we obtain explicit versions of  the lower estimates~\\eqref{ecolm} and~\\eqref{etriv}. In Section~\\ref{stenfourteen} we use all  previous results to bound the discriminant of a singular unit as  ${|\\Delta|<10^{15}}$. This bound is reduced to $10^{10}$ in Section~\\ref{sec:midrange} and to $3\\cdot10^5$ in Section~\\ref{slowrange}.  Finally, in Section~\\ref{sfinal} we show that the discriminant of a singular unit satisfies ${|\\Delta|>3\\cdot10^5}$. \n\n\\paragraph{Convention}\nIn this article we fix, once and for all, an embedding ${\\bar\\Q\\hookrightarrow \\C}$; this means that all algebraic numbers in this article are viewed as elements of~$\\C$.\n\n\\paragraph{Acknowledgments}\nYuri Bilu was partially supported by the University of Basel, the Fields Institute (Toronto), and the Xiamen University. Lars K\u00fchne was supported by the Max-Planck Institute for Mathematics, the Fields Institute, and the Swiss National Science Foundation through an Ambizione grant. We thank Ricardo Menares and Amalia Pizarro for many useful conversations, Florian Luca and Aleksandar Ivic for helpful suggestions, Bill Allombert and Karim Belabas for a \\textsf{PARI} tutorial, and Jean-Louis Nicolas and Cyril Mauvillain for helping to access Robin's thesis~\\cite{Ro83a}.  Finally, we thank both anonymous referees for encouraging reports and helpful suggestions. \n\n\n\\section{An estimate for \\texorpdfstring{${\\mathcal{C}}_\\varepsilon(\\Delta)$}{Ceps(Delta)}}\n\\label{sceps}\nLet~$\\Delta$ be a negative integer satisfying ${\\Delta\\equiv 0,1\\bmod 4}$ and  \n$$\n{\\mathcal{O}}_\\Delta=\\Z[(\\Delta+\\sqrt\\Delta)\/2] \n$$\nthe  imaginary quadratic order of discriminant~$\\Delta$. Then ${\\Delta=Df^2}$, where~$D$ is the discriminant of the imaginary quadratic field ${\\Q(\\sqrt\\Delta)}$  (the ``fundamental discriminant'') and ${f=[{\\mathcal{O}}_D:{\\mathcal{O}}_\\Delta]}$ is the conductor. We denote by ${\\mathcal{C}}(\\Delta)$ the class number  of the order~${\\mathcal{O}}_\\Delta$. \n\n\n\nUp to $\\C$-isomorphism there exist ${\\mathcal{C}}(\\Delta)$ elliptic curves with CM by~${\\mathcal{O}}_\\Delta$. The $j$-invariants of these curves are called \\textsl{singular moduli} of discriminant~$\\Delta$.\nThe singular moduli  of discriminant~$\\Delta$ form a full Galois orbit over~$\\Q$\nof cardinality ${\\mathcal{C}}(\\Delta)$, see \\cite[Proposition~13.2]{Cox}. \n\n\nLet~${\\mathcal{F}}$ be the standard fundamental domain in the Poincar\u00e9  plane,  that is, the open hyperbolic triangle with vertices ${\\zeta_3,\\zeta_6,i\\infty}$, together with the geodesics $[i,\\zeta_6]$ and ${[\\zeta_6,i\\infty)}$; here \n$$\n\\zeta_3=e^{2\\pi i\/3}=\\frac{-1+\\sqrt{-3}}2, \\quad \\zeta_6=e^{\\pi i\/3}=\\frac{1+\\sqrt{-3}}2. \n$$\nEvery singular modulus can be uniquely presented as  ${j(\\tau)}$, where ${\\tau\\in {\\mathcal{F}}}$. \n\n\nNow fix ${\\varepsilon\\in (0,1\/3]}$ and denote by ${\\mathcal{C}}_\\varepsilon(\\Delta)$ the number of singular moduli of discriminant~$\\Delta$ that can be presented as ${j(\\tau)}$ where ${\\tau\\in {\\mathcal{F}}}$ satisfies\n\\begin{equation}\n\\label{etauzeze}\n\\min \\{|\\tau-\\zeta_3|,|\\tau-\\zeta_6|\\}< \\varepsilon. \n\\end{equation}\nIn this section we bound this quantity.\n\n\nDefine the \\textsl{modified conductor}~${\\tilde f}$ by\n\\begin{equation}\n\\label{etilf}\n{\\tilde f}=\n\\begin{cases}\nf,& D\\equiv 1\\bmod 4,\\\\\n2f, &D\\equiv 0\\bmod 4. \n\\end{cases}\n\\end{equation}\nThen ${\\Delta\/{\\tilde f}^2}$ is a square-free integer. \n\n\n\n\n\\begin{theorem}\n\\label{thceps}\nFor ${\\varepsilon\\in (0,1\/3]}$ we have\n\\begin{equation}\n\\label{enewbound}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\le  F\\left( \\frac{16}3\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}}|\\Delta|^{1\/2}\\varepsilon^2+\\frac83|\\Delta|^{1\/2}\\varepsilon+ 8|\\Delta\/3|^{1\/4}\\sigma_0({\\tilde f})\\varepsilon+4\\right),\n\\end{equation}\nwhere  \n\\begin{equation}\n\\label{ecapitalf}\nF=F(\\Delta) = \\max \\bigl\\{2^{\\omega(a)}: a\\le |\\Delta|^{1\/2}\\bigr\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{corollary}\n\\label{cseps}\nIn the set-up of Theorem~\\ref{thceps} assume that ${|\\Delta|\\ge 10^{14}}$.  Then  \n\\begin{equation}\n\\label{enewboundsimple}\n{\\mathcal{C}}_\\varepsilon(\\Delta)  \\le  F\\left( 9.83|\\Delta|^{1\/2} \\varepsilon^2\\log\\log(|\\Delta|^{1\/2})+3.605|\\Delta|^{1\/2}\\varepsilon+ 4\\right).\n\\end{equation}\n\\end{corollary}\n\n\n\\subsection{Some lemmas}\nWe need some lemmas. For a prime number~$\\ell$ and a non-zero integer~$n$ we denote by ${\\mathrm{ord}}_\\ell(n)$ the $\\ell$-adic order of~$n$; that is, ${\\ell^{{\\mathrm{ord}}_\\ell(n)}\\,\\|\\,n}$. \n\n\\begin{lemma}\n\\label{lrootsmodle}\nLet~$\\ell$ be a prime number, ${e \\ge 1}$ an integer, and~$\\Delta$ a\nnon-zero integer with ${\\nu={\\mathrm{ord}}_\\ell\\Delta}$.\nThen the set of ${b\\in \\Z}$ satisfying \n${b^2\\equiv \\Delta\\bmod \\ell^e}$ \nis a union of at most~$2$ residue classes modulo ${\\ell^{e-\\lfloor\n    \\min\\{e,\\nu\\}\/2\\rfloor}}$ in all cases except when ${\\ell=2}$ and\n${e\\ge 3}$; in this latter case it is a union of most~$4$ such\nclasses. Finally, the set of $b$ equals a single residue class modulo ${\\ell^{e-\\lfloor\n    \\min\\{e,\\nu\\}\/2\\rfloor}}$\nif $\\nu\\ge e$. \n\\end{lemma}\n\\begin{proof}\nWe suppose first that ${\\nu=0}$, that is, ${\\ell\\nmid \\Delta}$.  In this case we have to count the number of elements in the multiplicative group $(\\Z\/\\ell^e\\Z)^\\times$ whose\nsquare is represented by~$\\Delta$. If ${\\ell\\ge 3}$ or ${\\ell^e\\in \\{2,4\\}}$, then $(\\Z\/\\ell^e\\Z)^\\times$ is a cyclic group.\nThen there are at most~$2$ square roots and this implies our claim. If ${\\ell=2}$ and  ${e \\ge 3}$, then ${(\\Z\/2^e\\Z)^\\times\\cong \\Z\/2\\Z\\times \\Z\/2^{e-2}\\Z}$, and there are at most~$4$ square roots, as desired.\n\nNow assume that  ${\\nu <e}$. Then ${{\\mathrm{ord}}_\\ell(b^2) = \\nu}$. So~$\\nu$ is even and we can write ${b = \\ell^{\\nu\/2}b'}$, where ${b' \\in \\Z}$ is coprime to~$\\ell$. Now ${\\Delta=\\ell^\\nu\\Delta'}$ \nwith ${\\Delta'\\in \\Z}$ coprime to~$\\ell$,  and ${(b')^2 \\equiv \\Delta'\\bmod \\ell^{e-\\nu}}$.  Above we already determined\nthat, depending on the value of $\\ell^{e-\\nu}$, the set of\npossible~$b'$ consists of either at most~$2$ or at most~$4$ classes\nmodulo ${\\ell^{e-\\nu}}$. Hence the set of possible ${b =\n  \\ell^{\\nu\/2}b'}$ consists of the same number of classes modulo\n${\\ell^{e-\\nu\/2}}$, as desired.\n\nTo prove the final claim  assume that ${\\nu \\ge e}$. In this case\n${b^2\\equiv \\Delta\\bmod \\ell^e}$ \nis equivalent to ${b\\equiv0\\bmod \\ell^{\\lceil e\/2\\rceil}}$. This means\nthat the set of suitable~$b$ consists of exactly one class  modulo\n${\\ell^{\\lceil e\/2\\rceil}=\\ell^{e-\\lfloor e\/2\\rfloor}}$.\n\\end{proof}\n\n\n\nWe say that ${d\\in \\Z}$ is a \\textsl{quadratic divisor} of ${n\\in \\Z}$ if ${d^2\\mid n}$. We denote by ${\\gcd_2(m,n)}$ the greatest common quadratic divisor of~$m$ and~$n$. \n\n\\begin{lemma}\n\\label{lrootsmoda}\nLet~$a$ be a positive integer and~$\\Delta$ a non-zero integer. Then the set of \n${b\\in \\Z}$ satisfying \n${b^2 \\equiv \\Delta \\bmod a}$\nconsists of at most   ${2^{\\omega(a\/\\gcd(a,\\Delta))+1}}$ residue classes modulo \n${a\/\\gcd_2(a,\\Delta)}$\n\\end{lemma}\n\\begin{proof}\n  For a prime power $\\ell^e$ \n  we only need the following simple consequence of \n  Lemma~\\ref{lrootsmodle} on the number of residue classes counted there.\nThis number is at\nmost $2^{\\omega(\\ell^e\/\\gcd(\\ell^e,\\Delta))}$ if $\\ell\\ge 3$\nand at most $2^{\\omega(\\ell^e\/\\gcd(\\ell^e,\\Delta))+1}$ for $\\ell=2$.\n  The current lemma follows from the Chinese\nRemainder Theorem.\n\\end{proof}\n\nThe  following lemma is trivial, but we state it here because it is our principal counting tool.  \n\n\\begin{lemma}\n\\label{ltrivi}\nLet~$\\alpha$ and~$\\beta$ be real numbers, ${\\alpha<\\beta}$, and~$m$ a positive integer. Then every residue class modulo~$m$ has at most ${(\\beta-\\alpha)\/m+1}$ elements in the interval ${[\\alpha,\\beta]}$.\n\\end{lemma}\n\n\n\nGiven a negative integer ${\\Delta\\equiv0,1\\bmod 4}$, denote by ${T=T_\\Delta}$ the set of triples of integers $(a,b,c)$ such that \n\\begin{equation}\n\\label{ekuh}\n\\begin{aligned}\n&\\gcd(a,b,c)=1, \\quad \\Delta=b^2-4ac,\\\\\n&\\text{either\\quad $-a < b \\le a < c$\\quad or\\quad $0 \\le b \\le a = c$.}\n\\end{aligned}\n\\end{equation}\nFor  ${(a,b,c)\\in T_\\Delta}$ we set\n$$\n\\tau(a,b,c)=\\frac{b+\\sqrt{\\Delta}}{2a}.\n$$\n\\begin{lemma}\n\\label{lgauss}\n\\begin{enumerate}[label={(\\roman*)}]\n\\item\n\\label{ifund}\nFor every ${(a,b,c)\\in T_\\Delta}$ the number \n$\\tau(a,b,c)$\nbelongs to the standard fundamental domain. \n\n\\item\n\\label{iadelta}\nFor ${(a,b,c)\\in T_\\Delta}$ we have ${0<a\\le |\\Delta\/3|^{1\/2}}$, the equality being possible only if ${\\Delta=-3}$ (and ${a=b=c=1}$). We also have ${c\\ge |\\Delta|^{1\/2}\/2}$. \n\n\\item\n\\label{icl}\nThe map\n${(a,b,c)\\mapsto j(\\tau(a,b,c))}$\ndefines a bijection from $T_\\Delta$ onto the set of $\\Q$-conjugates\nof $j(\\tau)$. \nIn particular, ${{\\mathcal{C}}(\\Delta)=|T_\\Delta|}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor item~\\ref{ifund} just note that~\\eqref{ekuh} implies the inequalities \n$$\n-\\frac12 <\\frac{b}{2a}\\le \\frac12, \\qquad \\frac{b^2+|\\Delta|}{4a^2}\\ge 1\n$$\nand that the second one becomes equality only when ${a=c}$, in which case  ${b\\ge 0}$. \nFor item~\\ref{iadelta}, \nsince ${|b|\\le a\\le c}$,  we have \n$$\n4c^2\\ge |\\Delta|=4ac-b^2\\ge 4a^2-a^2=3a^2. \n$$\nwith equality on the right only when ${a=|b|=c}$. Since ${\\gcd(a,b,c)=1}$, this is only possible when ${a=b=c=1}$ and ${\\Delta=-3}$. \n\nItem~\\ref{icl} is a combination of several classical results that can be found, for instance, in~\\cite{Cox}. See \\cite[Proposition~2.5]{BLP16} for more details. \n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lshort}\nLet ${\\varepsilon\\in (0,1\/3]}$ and let ${(a,b,c)\\in T_\\Delta}$ satisfy~\\eqref{etauzeze}. Then \n\\begin{align}\n\\label{eboundinga}\n\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}&<\\hphantom{|}a\\hphantom{|}\\le \\frac{|\\Delta|^{1\/2}}{\\sqrt3},\\\\\n\\label{eboundingb}\na(1-2\\varepsilon)&<|b|\\le a,\\\\\n\\label{eboundingc}\na&\\le \\hphantom{|}c\\hphantom{|}<a(1+{\\sqrt3}\\varepsilon+\\varepsilon^2)\n\\end{align}\n\\end{lemma}\nNote that~\\eqref{eboundinga} and~\\eqref{eboundingb} will be used already in Subsection~\\ref{ssproofthceps}, while~\\eqref{eboundingc} will be used only in Section~\\ref{slowrange}.\n\n\\begin{proof}\nFor ${\\tau\\in {\\mathcal{F}}}$ condition~\\eqref{etauzeze} implies that \n$$\n\\frac{\\sqrt3}2\\le\\Im\\tau<\\frac{\\sqrt3}2+\\varepsilon, \\qquad \n\\frac12-\\varepsilon<|\\Re\\tau|\\le \\frac12.\n$$\nApplying this for ${\\tau=\\tau(a,b,c)}$, we obtain~\\eqref{eboundinga} and~\\eqref{eboundingb}. To prove~\\eqref{eboundingc}, write\n$$\n4ac= |\\Delta|+b^2 < a^2(\\sqrt3+2\\varepsilon)^2+a^2=4a^2(1+\\sqrt3\\varepsilon+\\varepsilon^2), \n$$ \nand~\\eqref{eboundingc} follows. \n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thceps}}\n\\label{ssproofthceps}\n\nNote that, by definition, \n$$\n{\\mathcal{C}}_\\varepsilon(\\Delta) = \\#\\{\\text{$(a,b,c)\\in T_\\Delta$~: ${\\tau=\\tau(a,b,c)}$ satisfies~\\eqref{etauzeze}}\\}. \n$$\nSetting \n$$\nI=\\left(\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\frac{|\\Delta|^{1\/2}}{\\sqrt3}\\right], \n$$\nfor ${\\tau=\\tau(a,b,c)}$ with ${(a,b,c)\\in T_\\Delta}$ we may re-write~\\eqref{eboundinga} and~\\eqref{eboundingb} as   \n\\begin{align}\n\\label{eainibin}\na\\in I,\\qquad b\\in [-a,-a(1-2\\varepsilon))\\cup (a(1-2\\varepsilon),a]. \n\\end{align}\nSince~$c$ is uniquely determined for given~$a$,~$b$ and~$\\Delta$, it suffices to bound the number of  pairs $(a,b)$ of integers satisfying ${b^2\\equiv \\Delta\\bmod a}$ and~\\eqref{eainibin}. \n\nFor every fixed~$a$ there are at most\n${(4\\varepsilon\\gcd_2(a,\\Delta)+2) 2^{\\omega(a)+1}}$ suitable~$b$, as follows from Lemmas~\\ref{lrootsmoda} and~\\ref{ltrivi};\nindeed, ${\\omega(a\/\\gcd(a,\\Delta))\\le \\omega(a)}$. Hence\n\\begin{align}\n\\label{estim0}\n{\\mathcal{C}}_\\varepsilon(\\Delta) &\\le 8\\varepsilon\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta) 2^{\\omega(a)} + 4\\sum_{a\\in I\\cap\\Z}2^{\\omega(a)}\\\\ \n\\label{estim1}\n&\\le  8\\varepsilon F\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta) +4F\\#(I\\cap\\Z).\n\\end{align}\nTo estimate the sum, note that \n\\begin{equation}\n\\label{estim2}\n\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta)\\le \\sum_{d^2\\mid \\Delta}d\\cdot \\#(I\\cap d^2\\Z). \n\\end{equation}\nRecall that we defined in~\\eqref{etilf} the modified conductor~${\\tilde f}$. \nSince $\\Delta\/{\\tilde f}^2$  is a square-free integer, we have ${d^2\\mid \\Delta}$ if and only if ${d\\mid {\\tilde f}}$. Also, since~$I$ is of length \n$$\n|\\Delta|^{1\/2}\\left(\\frac1{\\sqrt3}-\\frac1{\\sqrt3+2\\varepsilon}\\right)<\\frac23|\\Delta|^{1\/2}\\varepsilon,\n$$ \nwe have, by Lemma~\\ref{ltrivi},\n$$\n\\#(I\\cap d^2\\Z) \\le \n\\begin{cases}\n\\frac23\\frac{|\\Delta|^{1\/2}}{d^2}\\varepsilon +1, & d \\le {|\\Delta\/3|^{1\/4}}, \\\\\n0,& d >{|\\Delta\/3|^{1\/4}}. \n\\end{cases}\n$$\nHence\n\\begin{align}\n\\sum_{d^2\\mid \\Delta}d\\cdot \\#(I\\cap d^2\\Z) &\\le \\sum_{\\genfrac{}{}{0pt}{}{d\\mid{\\tilde f}}{d\\le|\\Delta\/3|^{1\/4}}}d\\left(\\frac23\\frac{|\\Delta|^{1\/2}}{d^2}\\varepsilon+1\\right)\\nonumber\\\\\n&\\le \\frac23|\\Delta|^{1\/2}\\varepsilon\\sum_{d\\mid{\\tilde f}}d^{-1}+ \\sum_{\\genfrac{}{}{0pt}{}{d\\mid{\\tilde f}}{d\\le |\\Delta\/3|^{1\/4}}}d\\nonumber\\\\\n\\label{estim3}\n&\\le \\frac23\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}}|\\Delta|^{1\/2}\\varepsilon+ |\\Delta\/3|^{1\/4}\\sigma_0({\\tilde f}). \n\\end{align}\nFinally, Lemma~\\ref{ltrivi} implies that\n\\begin{equation}\n\\label{eintini}\n\\#(I\\cap\\Z)\\le \\frac23|\\Delta|^{1\/2}\\varepsilon+1. \n\\end{equation}\nPutting  the estimates~\\eqref{estim1},~\\eqref{estim2},~\\eqref{estim3} and~\\eqref{eintini} together, we obtain~\\eqref{enewbound}. \\qed\n\n\\subsection{Proof of Corollary~\\ref{cseps}}\n\nWe need to estimate  $\\sigma_0({\\tilde f})$ and $\\sigma_1({\\tilde f})$ in terms of~$|\\Delta|$.  The following lemma uses a simple estimate for $\\sigma_0(n)$ due to Nicolas and Robin~\\cite{NR83}. Much sharper  estimates can be found in Robin's thesis~\\cite{Ro83a}. \n\n\\begin{lemma}\n\\label{lsigzersimple}\nFor ${|\\Delta|\\ge 10^{14}}$ we have \n\\begin{align}\n\\label{euppersigzer}\n\\sigma_0({\\tilde f})&\\le |\\Delta|^{0.192},\\\\\n\\label{euppersigone}\n{\\sigma_1({\\tilde f})}\/{\\tilde f} &\\le 1.842\\log\\log(|\\Delta|^{1\/2}). \n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nFor proving~\\eqref{euppersigzer}  may assume that ${{\\tilde f}\\ge 16}$, otherwise there is nothing to prove. In~\\cite{NR83} it is proved that for ${n\\ge 3}$ we have\n$$\n\\frac{\\log\\sigma_0(n)}{\\log2} \\le 1.538\\frac{\\log n}{\\log\\log n}.\n$$\nThe function ${x\\mapsto (\\log x)\/(\\log\\log x)}$ is increasing for ${x\\ge 16}$. Since \n$$\n|\\Delta|\\ge 10^{14}, \\qquad 16\\le {\\tilde f}\\le |\\Delta|^{1\/2},\n$$ \nthis gives\n\\begin{align*}\n\\log\\sigma_0({\\tilde f}) \n&\\le 1.538\\log 2\\frac{\\log(|\\Delta|^{1\/2})}{\\log\\log(|\\Delta|^{1\/2})}\\\\\n&\\le \\frac{1.538}2\\log 2\\frac{\\log|\\Delta|}{\\log\\log(10^{7})}\\\\\n&<0.192\\log|\\Delta|,\n\\end{align*}\nas wanted. \n\nFor proving~\\eqref{euppersigone} we use the estimate \n${\\sigma_1(n)\\le 1.842n\\log\\log n}$\nwhich holds for ${n\\ge 121}$, see \\cite[Theorem~1.3]{AFJ07}. This\nproves~\\eqref{euppersigone} for ${{\\tilde f}\\ge 121}$. For ${{\\tilde f}\\le 120}$\none can check directly that  ${\\sigma_1({\\tilde f})\/{\\tilde f}\\le 3}$ so that inequality~\\eqref{euppersigone} is also\ntrue in this case.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cseps}]\nIf ${|\\Delta|\\ge 10^{14}}$ then Lemma~\\ref{lsigzersimple} implies that \n\\begin{align*}\n&8\\left|\\frac\\Delta3\\right|^{1\/4}\\sigma_0({\\tilde f}) \\le \\frac8{3^{1\/4}}|\\Delta|^{0.442}\\le \\frac8{3^{1\/4}\\cdot10^{0.812}}|\\Delta|^{1\/2}\\le 0.938|\\Delta|^{1\/2},\\\\\n&\\frac{16}3\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}} \\le \\frac{16}3\\cdot 1.842 \\log\\log(|\\Delta|^{1\/2})\\le9.83 \\log\\log(|\\Delta|^{1\/2}).\n\\end{align*}\nSubstituting all this to~\\eqref{enewbound}, we obtain~\\eqref{enewboundsimple}. \n\\end{proof}\n\n\n\n\\section{An upper bound for the height of a singular unit}\n\\label{supper}\n\nIn this section we obtain a fully explicit version of estimate~\\eqref{eupperheight}.\nWe use the notation ${\\mathcal{C}}(\\Delta)$,  ${\\mathcal{C}}_\\varepsilon(\\Delta)$, ${\\mathcal{F}}$, $\\zeta_3$, $\\zeta_6$ introduced in Section~\\ref{sceps}. \n\nLet~$\\alpha$ be a complex algebraic number of degree~$m$ whose minimal polynomial over~$\\Z$ is \n$$\nP(x)=a_mx^m+\\cdots+a_0=a_m(x-\\alpha_1)\\cdots (x-\\alpha_m) \\in \\Z[x].\n$$\nHere ${\\gcd(a_0,a_1, \\ldots, a_m)=1}$ and ${\\alpha_1, \\ldots, \\alpha_m\\in \\C}$ are the conjugates of~$\\alpha$ over~$\\Q$. Then the height of~$\\alpha$ is defined by\n$$\n{\\mathrm{h}}(\\alpha) =\\frac1m\\left(\\log|a_m|+\\sum_{k=1}^m \\log^+|\\alpha_k|\\right), \n$$ \nwhere ${\\log^+(\\cdot)=\\log\\max\\{1,\\cdot\\}}$. \nIf~$\\alpha$ is an algebraic integer then \n$$\n{\\mathrm{h}}(\\alpha) =\\frac1m\\sum_{k=1}^m \\log^+|\\alpha_k|. \n$$ \nIt is known that ${{\\mathrm{h}}(\\alpha)={\\mathrm{h}}(\\alpha^{-1})}$ when ${\\alpha\\ne 0}$. \n\n\\begin{theorem}\n\\label{theceps}\nLet~$\\alpha$ be a singular unit of discriminant~$\\Delta$, and~$\\varepsilon$ a real number satisfying ${0<\\varepsilon \\le 4\\cdot10^{-3}}$. \nThen \n\\begin{equation}\n\\label{eupperheightbis}\n{\\mathrm{h}}(\\alpha)\\le 3\\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta)}\\log|\\Delta|+3\\log(\\varepsilon^{-1})-10.66. \n\\end{equation}\n\\end{theorem}\n\nCombining this with Corollary~\\ref{cseps} and optimizing~$\\varepsilon$, we obtain the following consequence. \n\n\\begin{corollary}\n\\label{cupper}\nIn the set-up of Theorem~\\ref{theceps} assume that ${|\\Delta|\\ge 10^{14}}$. Then\n\\begin{align}\n\\label{eupdel}\n{\\mathrm{h}}(\\alpha)&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)}+3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.77,\n\\end{align}\nwhere ${A= F\\log|\\Delta|}$ and~$F$ is defined in~\\eqref{ecapitalf}.  \n\\end{corollary}\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{theceps}}\n\nWe start from some simple lemmas.\n\n\n\\begin{lemma}\n\\label{lanal}\nFor ${z\\in {\\mathcal{F}}}$ we have \n$$\n|j(z)|\\ge 42700\\bigl(\\min\\{|z-\\zeta_3|,|z-\\zeta_6|,4\\cdot10^{-3}\\}\\bigr)^3. \n$$\n\\end{lemma}\n\n\\begin{proof}\nThis is an easy modification  of Proposition~2.2 from~\\cite{BLP16}; just replace therein $10^{-3}$ by ${4\\cdot10^{-3}}$.   \n\\end{proof}\n\nIn the next lemma we use the notation $T_\\Delta$  and ${\\tau(a,b,c)}$ introduced before Lemma~\\ref{lgauss}. \n\n\\begin{lemma}\n\\label{lliouv}\nAssume that ${\\Delta\\ne -3}$. Let ${\\tau=\\tau(a,b,c)}$, where  ${(a,b,c)\\in T_\\Delta}$. Let~$\\zeta$ be one of the numbers~$\\zeta_3$ or~$\\zeta_6$. \nThen\n$$\n|\\tau-\\zeta|\\ge \\frac{\\sqrt3}{4|\\Delta|}. \n$$\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$$\n|\\tau-\\zeta|\\ge|\\Im\\tau-\\Im\\zeta|= \\left|\\frac{\\sqrt{|\\Delta|}}{2a}-\\frac{\\sqrt3}{2}\\right|= \\frac{\\bigl||\\Delta|-3a^2\\bigr|}{2a(\\sqrt{|\\Delta|}+a\\sqrt3)}.\n$$\nSince ${\\Delta\\ne -3}$ we have ${\\Delta\\ne -3a^2}$, see item~\\ref{iadelta} of Lemma~\\ref{lgauss}.  Hence\n$$\n|\\tau-\\zeta|\\ge \\frac{1}{2a(\\sqrt{|\\Delta|}+a\\sqrt3)}\\ge \\frac{\\sqrt3}{4|\\Delta|}, \n$$\nthe last inequality being again by item~\\ref{iadelta} of Lemma~\\ref{lgauss}. \n\\end{proof}\n\nNow we are ready to prove Theorem~\\ref{theceps}. \n\n\\begin{proof}[Proof of Theorem~\\ref{theceps}]\nLet ${\\alpha=\\alpha_1, \\alpha_2, \\ldots, \\alpha_m\\in \\C}$ be the conjugates of~$\\alpha$ over~$\\Q$. Then ${m={\\mathcal{C}}(\\Delta)}$ and ${\\alpha_1, \\ldots, \\alpha_m}$ is the full list of singular moduli of discriminant~$\\Delta$. Write them as ${j(\\tau_1), \\ldots, j(\\tau_m)}$, where ${\\tau_1, \\ldots, \\tau_m\\in {\\mathcal{F}}}$. \n\nSince~$\\alpha$ is a unit, we have \n$$\n{\\mathrm{h}}(\\alpha)={\\mathrm{h}}(\\alpha^{-1})  = \\frac1m \\sum_{k=1}^m\\log^+|\\alpha_k^{-1}|,\n$$\nHence \n{\\footnotesize\n\\begin{align}\n{\\mathrm{h}}(\\alpha) &= \\frac{1}{{\\mathcal{C}}(\\Delta)}\\sum_{k=1}^m\\log^+|j(\\tau_k)^{-1}|\\nonumber\\\\\n\\label{etwosums}\n&= \\frac{1}{{\\mathcal{C}}(\\Delta)}\\left(\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}< \\varepsilon}}+\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}\\ge \\varepsilon}}\\right)\\log^+|j(\\tau_k)^{-1}|.\n\\end{align}\n}%\nWe estimate each of the two sums separately.\n\nSince ${\\varepsilon\\le 4\\cdot10^{-3}}$, Lemma~\\ref{lanal} implies that each term in the second sum satisfies \n$$\n\\log^+|j(\\tau_k)^{-1}| \\le 3\\log(\\varepsilon^{-1})-\\log42700 \\le 3\\log(\\varepsilon^{-1})-10.66. \n$$\nHence\n$$\n\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}\\ge \\varepsilon}}\\log^+|j(\\tau_k)^{-1}|\\le ({\\mathcal{C}}(\\Delta)-{\\mathcal{C}}_\\varepsilon(\\Delta))\\bigl(3\\log(\\varepsilon^{-1})-10.66\\bigr). \n$$\nSince ${\\varepsilon\\le 4\\cdot10^{-3}}$ we have\n ${3\\log(\\varepsilon^{-1})>10.66}$, which implies that \n\\begin{equation}\n\\label{esecondsum}\n\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}\\ge \\varepsilon}}\\log^+|j(\\tau_k)^{-1}|\\le {\\mathcal{C}}(\\Delta)\\bigl(3\\log(\\varepsilon^{-1})-10.66\\bigr). \n\\end{equation}\nAs for the first sum, Lemmas~\\ref{lanal} and~\\ref{lliouv} imply that each term in this sum satisfies\n$$\n\\log^+|j(\\tau_k)^{-1}| \\le \\max\\left\\{0, 3\\log\\frac{4|\\Delta|}{\\sqrt3}-\\log42700\\right\\} \\le 3\\log|\\Delta|.  \n$$ \nNote that we may use here Lemma~3.4 because the only singular modulus of discriminant $-3$ is~$0$, which is not a unit. \n\nSince the first sum has ${\\mathcal{C}}_\\varepsilon(\\Delta)$ terms, this implies the estimate \n\\begin{equation}\n\\label{efirstsum}\n\\sum_{\\genfrac{}{}{0pt}{}{1\\le k\\le m}{\\min\\{|\\tau_k-\\zeta_3|, |\\tau_k-\\zeta_6|\\}< \\varepsilon}}\\log^+|j(\\tau_k)^{-1}|\\le 3{\\mathcal{C}}_\\varepsilon(\\Delta)\\log|\\Delta|. \n\\end{equation}\nSubstituting~\\eqref{esecondsum} and~\\eqref{efirstsum} into~\\eqref{etwosums}, we obtain~\\eqref{eupperheightbis}.\n\\end{proof}\n\n\n\\subsection{Proof of Corollary~\\ref{cupper}}\n\nTo prove the corollary we need a lower bound for the quantity~$F$ defined in Theorem~\\ref{thceps} and an upper bound for the class number ${\\mathcal{C}}(\\Delta)$.  \n\n\\begin{lemma}\n\\label{lf}\nAssume that ${|\\Delta|\\ge 10^{14}}$. Then   ${F\\ge |\\Delta|^{0.34\/\\log\\log(|\\Delta|^{1\/2})}}$ and ${F\\ge 18.54\\log\\log(|\\Delta|^{1\/2})}$.  \n\\end{lemma}\n\n\\begin{proof}\nDefine, as usual \n\\begin{equation}\n\\label{ethetapi}\n\\vartheta(x)=\\sum_{p\\le x}\\log p, \\qquad \\pi(x)=\\sum_{p\\le x}1. \n\\end{equation}\nThen \n\\begin{align}\n\\vartheta(x) & \\le 1.017 x &&(x>0), \\nonumber\\\\\n\\label{elowerpi}\n\\pi(x) &\\ge \\frac{x}{\\log x} && (x\\ge 17),\n\\end{align} \nsee \\cite{RS62}, Theorem~9 on page~71 and Corollary~1 after Theorem~2 on page~69. Estimate~\\eqref{elowerpi} implies that\n\\begin{equation}\n\\label{elowerpibis}\n\\pi(x) \\ge 0.99995\\frac{x}{\\log x}  \\qquad (x\\ge 13).\n\\end{equation}\nSetting here \n$$\nx=\\frac{\\log(|\\Delta|^{1\/2})}{1.017} , \\qquad N =\\prod_{p\\le x}p,\n$$\nwe obtain ${N\\le |\\Delta|^{1\/2}}$ and \n$$\n\\omega(N) =\\pi(x) \\ge \\frac{0.99995\\log(|\\Delta|^{1\/2})}{1.017\\log\\log(|\\Delta|^{1\/2})}. \n$$\nNote that ${x\\ge \\bigl(\\log(10^{7})\\bigr)\/1.017>15}$, so we are allowed to use~\\eqref{elowerpibis}. \nWe obtain \n$$\nF\\ge 2^{\\omega(N)} \\ge |\\Delta|^{\\frac{0.99995\\log2}{2\\cdot1.017\\log\\log(|\\Delta|^{1\/2})}}\\ge |\\Delta|^{0.34\/\\log\\log(|\\Delta|^{1\/2})},\n$$\nproving the first estimate.\n\n\nTo prove the second estimate, we deduce from the first estimate that \n\\begin{equation}\n\\label{elog-log}\n\\log F-\\log\\log\\log(|\\Delta|^{1\/2})\\ge 0.68 \\frac{u}{\\log u}-\\log\\log u,\n\\end{equation} \nwhere we set ${u=\\log(|\\Delta|^{1\/2})}$. The right-hand side of~\\eqref{elog-log}, viewed as a function in~$u$, is increasing for ${u\\ge \\log(10^{7})}$. Hence \n$$\n\\log F-\\log\\log\\log(|\\Delta|^{1\/2})\\ge 0.68 \\frac{\\log(10^{7})}{\\log \\log(10^{7})}-\\log\\log \\log(10^{7}) \\ge 2.92,\n$$\nand ${F\\ge e^{2.92}\\log\\log(|\\Delta|^{1\/2}) \\ge 18.54 \\log\\log(|\\Delta|^{1\/2})}$. \n\\end{proof}\n\n\\begin{lemma}\n\\label{lcd}\nFor ${\\Delta\\ne -3,-4}$ we have \n$$\n{\\mathcal{C}}(\\Delta) \\le \\pi^{-1}|\\Delta|^{1\/2}(2+\\log|\\Delta|).\n$$ \n\\end{lemma}\n\n\\begin{proof}\nThis follows from Theorems~10.1 and~14.3 in \\cite[Chapter~12]{Hu82}. Note that in~\\cite{Hu82} the right-hand side has an extra factor $\\omega\/2$, where~$\\omega$ is the number of roots of unity in the imaginary quadratic order of discriminant~$\\Delta$. Since we assume that ${\\Delta \\ne -3,-4 }$, we have ${\\omega=2}$, so we may omit this factor. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cupper}]\nSubstituting the estimate for ${\\mathcal{C}}_\\varepsilon(\\Delta)$ from~\\eqref{enewboundsimple} into~\\eqref{eupperheightbis}, we obtain the estimate\n\\begin{align*}\n{\\mathrm{h}}(\\alpha)&\\le 3A\\frac{9.83|\\Delta|^{1\/2} \\varepsilon^2\\log\\log(|\\Delta|^{1\/2})+3.605|\\Delta|^{1\/2}\\varepsilon+ 4}{{\\mathcal{C}}(\\Delta)}\n\\\\&\\hphantom{\\le}\n+3\\log(\\varepsilon^{-1})-10.66\n\\end{align*}\nwith ${A=F\\log|\\Delta|}$. Specifying \n$$\n\\varepsilon=0.27\\frac{{\\mathcal{C}}(\\Delta)}{A|\\Delta|^{1\/2}}\n$$\n(this is a nearly optimal value, and it satisfies ${\\varepsilon\\le 4\\cdot10^{-3}}$ as verified below), we obtain, using Lemmas~\\ref{lf} and~\\ref{lcd},\n\\begin{align*}\n{\\mathrm{h}}(\\alpha)&\\le  3\\cdot 9.83 \\cdot (0.27)^2\\frac{\\log\\log(|\\Delta|^{1\/2})}F\\frac{{\\mathcal{C}}(\\Delta)}{|\\Delta|^{1\/2}\\log|\\Delta|}+3\\cdot3.605\\cdot0.27\\\\ \n&\\hphantom{\\le}+\\frac{12A}{{\\mathcal{C}}(\\Delta)} +3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3\\log0.27-10.66\\\\\n&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)} +3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}+ \\frac{3\\cdot 9.83 \\cdot (0.27)^2\\cdot0.34}{18.54}+3\\cdot3.605\\cdot0.27\\\\ \n&\\hphantom{\\le}-3\\log0.27-10.66\\\\\n&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)} +3\\log\\frac{A|\\Delta|^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.77,\n\\end{align*}\nas wanted. \n\n \nWe only have to verify that ${\\varepsilon\\le 4\\cdot10^{-3}}$. We have\n${F\\ge 256}$ when ${|\\Delta|\\ge 10^{14}}$. Using Lemma~\\ref{lcd}, we obtain \n\\begin{align*}\n\\varepsilon=\\frac{0.27{\\mathcal{C}}(\\Delta)}{|\\Delta|^{1\/2}\\log |\\Delta|}\\frac1F\n\\le 0.27\\pi^{-1}\\frac{2+\\log(10^{14})}{\\log(10^{14})}\\cdot \\frac1{256}\n<4\\cdot 10^{-4}.\n\\end{align*}\nThe proof is complete.\n\\end{proof}\n\n\n\n\n\\section{Lower bounds for the height of a singular modulus}\n\\label{slower}\n\nNow we establish explicit lower bounds of the form~\\eqref{ecolm} and~\\eqref{etriv}. \n\n\\subsection{The ``easy'' bound}\n\nWe start by proving a bound of the form~\\eqref{etriv}. \n\n\\begin{proposition}\n\\label{ptriv}\nLet~$\\alpha$ be a singular modulus of discriminant~$\\Delta$. \nAssume that ${|\\Delta|\\ge 16}$. Then \n\\begin{equation}\n\\label{etrivbis}\n{\\mathrm{h}}(\\alpha) \\ge  \\frac{\\pi|\\Delta|^{1\/2}-0.01}{{\\mathcal{C}}(\\Delta)}.\n\\end{equation}\n\\end{proposition}\n\n\nWe need a simple lemma. \n\n\\begin{lemma}\n\\label{lttsn}\nFor ${z\\in {\\mathcal{F}}}$ with imaginary part~$y$ we have \n$$\n\\bigl||j(z)|-e^{2\\pi y}\\bigr|\\le 2079.\n$$ \nIf ${y\\ge 2}$ then we also have ${|j(z)|\\ge 0.992 e^{2\\pi y}}$. \n\\end{lemma}\n\n\\begin{proof}\nThe first statement is Lemma~1 of~\\cite{BMZ13}, and the second one is an immediate consequence. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{ptriv}]\nOne of the conjugates of~$\\alpha$ over~$\\Q$ is equal to ${j((b+\\sqrt\\Delta)\/2)}$, with ${b=1}$ for~$\\Delta$ odd, and ${b=0}$ for~$\\Delta$ even;  it corresponds to the element ${(1,b,(-\\Delta+b^2)\/4)}$ of the set $T_\\Delta$. \nHence\n$$\n{\\mathrm{h}}(\\alpha) \\ge  \\frac{\\log|j((b+\\sqrt\\Delta)\/2)|}{{\\mathcal{C}}(\\Delta)}.\n$$\nUsing Lemma~\\ref{lttsn}, we obtain\n$$\n\\log|j((b+\\sqrt\\Delta)\/2)|\\ge \\pi|\\Delta|^{1\/2}+\\log 0.992 \\ge \\pi|\\Delta|^{1\/2}-0.01.\n$$\nWhence the result. \n\\end{proof}\n\n\\subsection{The ``hard'' bound}\n\n\\label{sshard}\n\nWe are left with  bound~\\eqref{ecolm}. We are going to prove the following. \n\n\\begin{proposition}\n  \\label{phlb2}\nLet~$\\alpha$ be a singular modulus of discriminant~$\\Delta$.\nThen\n\\begin{equation}\n\\label{ecolmbis}\n{\\mathrm{h}}(\\alpha) \\ge \\frac{3}{\\sqrt{5}}\\log|\\Delta| -9.79. \n\\end{equation}\n\\end{proposition}\n\n\nThe proof of Proposition~\\ref{phlb2} relies on the\nfact that it is possible to evaluate the Faltings height of an elliptic curve\nwith complex multiplication precisely, due to the work of Colmez~\\cite{Colmez} and Nakkajima-Taguchi~\\cite{NT}; for an exact statement see  \\cite[Lemma 4.1]{Ha10}. \n\nLet~$E$ be an elliptic curve with CM by an order of discriminant~$\\Delta$.  \nWe let ${\\mathrm{h}}_F(E)$ denote the stable Faltings height of~$E$ (using Deligne's normalization~\\cite{De85}). \nThe above-mentioned explicit formula for ${\\mathrm{h}}_F(E)$ is used in~\\cite{HJM:6} to obtain the lower bound  \n\\begin{equation}\n\\label{efaltlowerold}\n{\\mathrm{h}}_F(E) \\ge \\frac{1}{4\\sqrt5} \\log|\\Delta| - 5.93,\n\\end{equation}\nsee Lemma 14(ii) therein. Unfortunately, this bound is numerically too weak for our purposes.\n\nProposition~\\ref{phlb2} will be deduced from the following numerical refinement of~\\eqref{efaltlowerold}.\n\n\\begin{proposition}\n\\label{pfaltlower}\nLet~$E$ be an elliptic curve with CM by an order of discriminant~$\\Delta$.\nThen \n\\begin{equation}\n\\label{efaltlowernew}\n{\\mathrm{h}}_F(E) \\ge \\frac{1}{4\\sqrt5} \\log|\\Delta| -\\gamma -\\frac{\\log(2\\pi)}{2}-\\left(\\frac1{2\\sqrt5}-\\frac16\\right)\\log2,\n\\end{equation}\nwhere ${\\gamma=0.57721\\ldots}$ is the Euler constant. \n\\end{proposition}\n\n\nLet us first show how Proposition~\\ref{pfaltlower} implies Proposition~\\ref{phlb2}. \n\n\n\\begin{proof}[Proof of Proposition~\\ref{phlb2} (assuming Proposition~\\ref{pfaltlower})]\nLet~$E$ be an elliptic curve with ${j(E)=\\alpha}$. \nWe only need to relate ${\\mathrm{h}}_F(E)$ to ${\\mathrm{h}}(j(E))$. For this purpose we use Lemma~7.9 of Gaudron and R\\'emond~\\cite{GaudronRemond:periodes}\\footnote{The reader should be warned that our ${\\mathrm{h}}_F(E)$ is denoted ${\\mathrm{h}}(E)$ in~\\cite{GaudronRemond:periodes}.}.\nIn our\nnotation  they\nshow that \n\\begin{equation}\n\\label{egr}\n{\\mathrm{h}}_F(E)\\le {\\mathrm{h}}(j(E))\/12 - 0.72 \n\\end{equation}\nA quick calculation yields  our\nclaim. \n\\end{proof}\n\n\n\nTo prove Proposition~\\ref{pfaltlower} we need a technical lemma.\nSet \n\\begin{equation}\n\\label{elambda}\n\\lambda=\\frac12-\\frac1{2\\sqrt5},\n\\end{equation}\nand define the additive arithmetical functions $\\beta(n)$ and $\\delta(n)$ by  \n\\begin{equation}\n\\label{ebetadelta}\n\\beta(p^k) =\\frac{\\log p}{p+1}\\frac{1-p^{-k}}{1-p^{-1}}, \\quad \\beta(n)=\\sum_{p^k\\|n}\\beta(p^k), \\quad \\delta(n)= \\lambda\\log n -\\beta(n).\n\\end{equation}\n\n\\begin{lemma}\n\\label{ltechn} \nFor every positive integer~$n$ we have \n$$\n\\delta(n) \\ge\\delta(2)= \\left(\\frac16-\\frac1{2\\sqrt5}\\right)\\log2. \n$$\n\\end{lemma}\n\n\\begin{proof}\nSince ${1\/3>\\lambda>1\/4}$, we have ${\\delta(2)<0}$ and  ${\\delta(p)>0}$ for all primes ${p\\ge 3}$. Also, for ${k\\ge 1}$ and any prime~$p$ we have \n$$\n\\delta(p^{k+1})-\\delta(p^k)= \\left(\\lambda-\\frac1{p^k(p+1)}\\right)\\log p >0.\n$$\nSince ${\\delta(4)>0}$, this proves that ${\\delta(p^k)>0}$ for every prime power ${p^k\\ne 2}$, whence the result. \n\\end{proof}\n\nProposition~\\ref{pfaltlower} is an immediate consequence of Lemma~\\ref{ltechn} and the following statement.  \n\n\\begin{proposition}\n\\label{pfaltlowerf}\nIn the set-up of Proposition~\\ref{pfaltlower} we have\n\\begin{equation}\n\\label{efaltlowernewf}\n{\\mathrm{h}}_F(E) \\ge \\frac{1}{4\\sqrt5} \\log|\\Delta| + \\lambda \\log f- \\beta(f) -\\gamma -\\frac{\\log(2\\pi)}{2}.\n\\end{equation}\n\\end{proposition}\nSince \n$$\n\\lambda \\log f- \\beta(f)\\ge-\\left(\\frac1{2\\sqrt5}-\\frac16\\right)\\log2\n$$\nby Lemma~\\ref{ltechn}, this implies Proposition~\\ref{pfaltlower}. \n\n\\begin{proof}[Proof of Proposition~\\ref{pfaltlowerf}]\nWrite ${\\Delta=Df^2}$ with~$D$ the fundamental discriminant and~$f$ the conductor. Define \n$$\ne_f(p) = \\frac{1-\\chi(p)}{p-\\chi(p)}\\frac{1-p^{-{\\mathrm{ord}}_p(f)}}{1-p^{-1}}, \\qquad c(f) = \\frac 12 \\left(\\sum_{p\\mid f} e_f(p)\\log p\\right),\n$$\nwhere ${\\chi(\\cdot)=(D\/\\cdot)}$ is Kronecker's symbol.\n\n\nIn the proof of Lemma~14 of~\\cite{HJM:6}\\footnote{Note that our~$D$ is written~$\\Delta$ in~\\cite{HJM:6}.},\nthe stable Faltings height of~$E$  is estimated as \n\\begin{align*}\n{\\mathrm{h}}_F(E) &\\ge \\frac{1}{4\\sqrt5}\\log|D| + \\frac 12 \\log f\n-c(f)\n-\\gamma -\\frac{\\log(2\\pi)}{2},\\\\\n&=\\frac{1}{4\\sqrt5}\\log|\\Delta| + \\lambda \\log f\n-c(f)\n-\\gamma -\\frac{\\log(2\\pi)}{2}.\n\\end{align*}\nThus, to establish~\\eqref{efaltlowernewf}, we only have to prove that ${c(f)\\le \\beta(f)}$. \nWe have \n$$\n\\frac{1-\\chi(p)}{p-\\chi(p)} = \n\\begin{cases}\n0, &\\chi(p)=1,\\\\\n1\/p, &\\chi(p)=0,\\\\\n 2\/(p+1), &\\chi(p)=-1. \n\\end{cases}\n$$\nHence\n$$\n\\frac{1-\\chi(p)}{p-\\chi(p)} \\le \\frac2{p+1}\n$$\nin any case. This implies that ${c(f)\\le \\beta(f)}$.\nThe proposition is proved. \n\\end{proof}\n\n\n\\section{The estimate \\texorpdfstring{${|\\Delta|< 10^{15}}$}{|Delta|<10to15}}\n\\label{stenfourteen}\n\n\n\nIn this section we obtain the first explicit upper bound for the\ndiscriminant of a singular unit. \n\n\n\\begin{theorem}\n  \\label{thhighrange}\nLet~$\\Delta$  be the discriminant of a singular unit. Then\n  ${|\\Delta|<10^{15}}$. \n\\end{theorem}\n\n{\\sloppy\n\nThroughout this section~$\\Delta$ is the discriminant of a singular unit~$\\alpha$, and  we assume that  ${X=|\\Delta|\\ge 10^{15}}$, as otherwise there is nothing to prove. \nOur principal tools will be the upper estimate~\\eqref{eupdel}\nand  the lower estimates~\\eqref{etrivbis},~\\eqref{ecolmbis}. We reproduce them here for convenience:\n\\begin{align}\n\\label{eupx}\n{\\mathrm{h}}(\\alpha)&\\le \\frac{12A}{{\\mathcal{C}}(\\Delta)}+3\\log\\frac{AX^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.77,\\\\\n\\label{etrivx}\n{\\mathrm{h}}(\\alpha)&\\ge  \\frac{\\pi X^{1\/2}-0.01}{{\\mathcal{C}}(\\Delta)},\\\\\n\\label{ecolmx}\n{\\mathrm{h}}(\\alpha)&\\ge \\frac{3}{\\sqrt5}\\log X-9.79.\n\\end{align}\nNote that our assumption ${X\\ge 10^{15}}$ implies that the right-hand side of~\\eqref{ecolmx} is positive.\n\n}\n\n\n\\subsection{The main inequality}\n\\label{ssinequality}\n\nRecall that ${A= F\\log X}$. \nMinding~$0.01$ in~\\eqref{etrivx} we deduce from~\\eqref{eupx},~\\eqref{etrivx} and~\\eqref{ecolmx} the inequality\n$$\n\\frac{12A}{{\\mathcal{C}}(\\Delta)}+3\\log\\frac{AX^{1\/2}}{{\\mathcal{C}}(\\Delta)}-3.76 \\ge \\max\\left\\{\\frac{\\pi X^{1\/2}}{{\\mathcal{C}}(\\Delta)}, \\frac{3}{\\sqrt5}\\log X-9.78\\right\\}. \n$$\nDenoting\n\\begin{equation}\n\\label{ey}\nY=\\max\\left\\{\\frac{\\pi X^{1\/2}}{{\\mathcal{C}}(\\Delta)}, \\frac{3}{\\sqrt5}\\log X-9.78\\right\\}, \n\\end{equation}\nwe re-write this as \n\\begin{equation}\n\\label{enotyet}\n\\frac{12A\/{\\mathcal{C}}(\\Delta)}{Y}+\\frac{3\\log A-3.76}Y+ \\frac{\\log(X^{1\/2}\/{\\mathcal{C}}(\\Delta))}{Y} \\ge 1. \n\\end{equation}\nNote that ${3\\log A-3.76>0}$, because ${A\\ge  \\log X\\ge \\log(10^{15}) >30}$. Hence we may replace~$Y$ by ${\\frac{3}{\\sqrt5}\\log X-9.78}$ in the middle term of the left-hand side in~\\eqref{enotyet}. Similarly, in the first term we may replace~$Y$ by ${\\pi X^{1\/2}\/{\\mathcal{C}}(\\Delta)}$, and in the third term we may replace ${X^{1\/2}\/{\\mathcal{C}}(\\Delta)}$ by ${\\pi^{-1}Y}$. We obtain \n\\begin{equation}\n\\label{einequality}\n12\\pi^{-1} AX^{-1\/2}+ \\frac{3\\log A-3.76}{\\frac{3}{\\sqrt5}\\log X-9.78} + 3\\frac{\\log(\\pi^{-1}Y)}{Y}\\ge 1. \n\\end{equation}\n\n\nTo show that~\\eqref{einequality} is not possible for ${X\\ge 10^{15}}$, we will bound from above each of the three terms in its left-hand side.  To begin with, we bound~$A$. \n\n\n\\subsection{Bounding~$F$ and~$A$}\n\nRecall that ${F=\\max\\{2^{\\omega(a)}: a\\le X^{1\/2}\\}}$ and ${A=F\\log X}$. \n\nLet  \n${N_1 = 2\\cdot3\\cdot5\\cdots 1129}$  be the product of the first $189$ prime numbers. Define the real number~$c_1$ from \n\\begin{align*}\n\\omega(N_1)&=\\frac{\\log N_1}{\\log\\log N_1 -c_1}.\n\\end{align*}\nA calculation shows that ${c_1 < 1.1713142}$. \nRobin  \\cite[Th\u00e9or\u00e8me~13]{Ro83} proved that\n\\begin{equation*}\n\\omega(n) \\le \\frac{\\log n}{\\log\\log n-c_1} \n\\end{equation*}\nfor ${n\\ge 26}$. \nThis implies that \n\\begin{align}\n\\label{eboundf}\n\\frac{\\log F}{\\log 2} &\\le \\frac12\\frac{\\log X}{\\log\\log X-c_1-\\log2} ,\\\\\n\\label{ebounda}\n\\log A &\\le \\frac{\\log2}2\\frac{\\log X}{\\log\\log X-c_1-\\log2}+\\log\\log X. \n\\end{align}\nIndeed, the function \n$$\ng(x)= \\frac{\\log x}{\\log\\log x-c_1}\n$$\nis strictly increasing for ${x\\ge 6500}$ and ${g(6500)>8}$. \nIf ${a\\le X^{1\/2}}$ then either ${a\\le 6500}$ in which case \n${\\omega(a) \\le 5<g(6500)<g(X^{1\/2})}$\n(recall that ${X\\ge 10^{15}}$), or ${6500<a\\le X^{1\/2}}$, in which case \n${\\omega(a) \\le g(a)\\le g(X^{1\/2})}$.\nThus, in any case we have \n$$\n\\omega(a) \\le g(X^{1\/2})= \\frac12\\frac{\\log X}{\\log\\log X-c_1-\\log2}, \n$$\nwhich proves~\\eqref{eboundf}. The estimate~\\eqref{ebounda} is an immediate consequence of~\\eqref{eboundf}.  \n\n\n\\subsection{Bounding the first term in~\\eqref{einequality}}\nUsing~\\eqref{ebounda}, we estimate\n$$\n\\frac{\\log (AX^{-1\/2})}{\\log X}\\le u_0(X), \n$$\nwhere\n\\begin{align*}\nu_0(x)&= \\frac{\\log2}2\\frac{1 }{\\log\\log x-c_1-\\log2}+\\frac{\\log\\log x}{\\log x}-\\frac12. \n\\end{align*}\nThe function $u_0(x)$ is decreasing for ${x\\ge 10^{10}}$. Hence \nfor ${X\\ge 10^{15}}$ we have \n$$\n\\frac{\\log (AX^{-1\/2})}{\\log X}\\le u_0(10^{15}) <-0.1908. \n$$\nThis proves the estimate \n\\begin{equation*}\nAX^{-1\/2}<  10^{15\\cdot(-0.1908)}<0.0014\n\\end{equation*}\nfor ${X\\ge 10^{15}}$. \n\n\n\\subsection{Bounding the second term in~\\eqref{einequality}}\nUsing~\\eqref{ebounda}, we estimate \n$$\n\\frac{3\\log A-3.76}{\\frac{3}{\\sqrt5}\\log X-9.78}\\le u_1(X)u_2(X), \n$$\nwhere\n\\begin{align*}\nu_1(x) &= \\frac{3\\log2}2\\frac{1}{\\log\\log x -c_1-\\log2}+\\frac{3\\log\\log x-3.76}{\\log x},\\\\\nu_2(x)& = \\left(\\frac{3}{\\sqrt5}-\\frac{9.78}{\\log x}\\right)^{-1}. \n\\end{align*}\nBoth functions $u_1(x)$ and $u_2(x)$ are decreasing for ${x\\ge 10^{10}}$. Hence, for ${X\\ge 10^{15}}$ we have\n\\begin{equation*}\n\\frac{3\\log A-3.76}{\\frac{3}{\\sqrt5}\\log X-9.78}\\le u_1(10^{15})u_2(10^{15}) <0.7734. \n\\end{equation*}\n\n\n\\subsection{Bounding the third term in~\\eqref{einequality}}\n\nThe function ${x\\mapsto (\\log x)\/x}$ is decreasing for ${x\\ge e}$. Since for ${X\\ge 10^{15}}$ we have \n$$\n\\pi^{-1}Y\\ge \\pi^{-1}\\left(\\frac{3}{\\sqrt5}\\log X-9.78\\right)\\ge e,\n$$\nwe have, for ${X\\ge 10^{15}}$, the estimate \n$$\n\\frac{\\log(\\pi^{-1}Y)}{Y} \\le u_3(X), \n$$\nwhere \n$$\nu_3(x) = \\frac{\\log\\left(\\pi^{-1}\\bigl(\\frac{3}{\\sqrt5}\\log x-9.78\\bigr)\\right)}{\\frac{3}{\\sqrt5}\\log x-9.78}. \n$$\nMoreover, the function $u_3(x)$ is decreasing for ${x\\ge 10^{15}}$, which implies that \n\\begin{equation*}\n\\frac{\\log(\\pi^{-1}Y)}{Y}\\le u_3(X) \\le u_3(10^{15}) < 0.0672\n \\end{equation*}\nfor ${X\\ge 10^{15}}$. \n\n\\subsection{Summing up}\n\nNow, when ${X\\ge 10^{15}}$, we can combine the above estimates and bound the left-hand side of~\\eqref{einequality} by\n$$\n12\\pi^{-1}\\cdot0.0014 + 0.7734 + 3 \\cdot 0.0672 <0.981.\n$$\nHence, for ${X\\ge 10^{15}}$ we cannot have~\\eqref{einequality}. This contradiction completes the proof of Theorem~\\ref{thhighrange}. \n\n\n\n\n\n\\section{Handling the mid-range \\texorpdfstring{$10^{10}\\le |\\Delta|< 10^{15}$}{10to10<Delta<10to15}}\n\\label{sec:midrange}\n\n\nIn this section we rule out the existence of singular units with discriminants in the\nmid-range ${[10^{10}, 10^{15})}$, improving thereby the bound from the previous section.\n\n\n\\begin{theorem}\n  \\label{thmidrange}\nLet~$\\Delta$  be the discriminant of a singular unit. Then\n  ${|\\Delta|\\notin [10^{10},10^{15})}$. \n\\end{theorem}\n\n\n\nIn Section~\\ref{sceps} we estimate trivially ${2^{\\omega(a)}\\le F}$. One might expect to do better by estimating the average order rather than the maximal order of the arithmetical function ${2^{\\omega(n)}}$. This is accomplished in Subsection~\\ref{ssdelange} and  allows us to obtain, in Subsection~\\ref{ssnewceps}, a new bound for ${{\\mathcal{C}}_\\varepsilon(\\Delta)}$ in the range ${10^{10}\\le |\\Delta|< 10^{15}}$. Using this, Theorem~\\ref{thmidrange} is proved in Subsection~\\ref{ssproofmidrange} by  an argument similar to the proof of Theorem~\\ref{thhighrange}. \n\nThroughout this section~$n$ denotes a positive integer. \n\n\\subsection{Average order of the function $2^{\\omega(n)}$ on subintervals of ${[0,2\\cdot10^7]}$}\n\\label{ssdelange}\nFor a positive  real number~$x$ set \n$$\nS(x)= \\sum_{n\\le x} 2^{\\omega(n)}.\n$$\nWe define $S(0)=0$.\nAs Theorem II.6.1 from~\\cite{Te15} suggests, the function $S(x)$ can be well approximated by the function\n$$\ng(x)= \\lambda_0x\\log x+\\lambda_1 x,\n$$\nwhere \n\\begin{align*}\n\\lambda_0&= \\zeta(2)^{-1}=0.607927101854026\\dots,\\\\%62866327677925837\\dots,\\\\ \n\\lambda_1 &= -2\\frac{\\zeta^{\\prime}(2)}{\\zeta(2)^{2}}+\\frac{2\\gamma-1}{\\zeta(2)}=0.786872460166245\\dots\n\\end{align*}\nand~$\\gamma$ is the Euler constant. The function~$g$ is increasing on\n$[1,\\infty)$. \n\n\nAs already mentioned in the introduction, the error term ${|S(x)-g(x)|}$ can be estimated by the Selberg-Delange method~\\cite[Chapter~II.5]{Te15}, but on our\nlimited range it is more advantageous to obtain an optimal error term by a computer-assisted\ncalculation.\n\n\\begin{proposition}\n\\label{proposition::selbergdelange}\nFor ${2\\le x \\le 2\\cdot 10^7}$ we have \n\\begin{equation}\n\\label{esqrt}\ng(x)- 1.010x^{1\/2} \\le S(x) \\le g(x)+ 0.712x^{1\/2}, \n\\end{equation}\nand for ${4\\cdot10^4\\le x\\le 2\\cdot 10^7}$ we have \n\\begin{equation}\n\\label{esqrtlog}\ng(x)- 2.267\\frac{x^{1\/2}}{\\log x} \\le S(x) \\le g(x)+ 2.598\\frac{x^{1\/2}}{\\log x}. \n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}\nSet\n\\begin{align*}\nc_1&=\\max_{2\\le n\\le 2\\cdot 10^7}\\frac{S(n)-g(n)}{\\sqrt n}, \\\\\nc_2&=\\max_{2\\le n\\le 2\\cdot 10^7}\\frac{g(n+1)-S(n)}{\\sqrt n},\\\\\nc_3&=\\max_{4\\cdot10^4\\le n\\le 2\\cdot 10^7}\\frac{(S(n)-g(n))\\log n}{\\sqrt n}, \\\\\nc_4&=\\max_{4\\cdot10^4\\le n\\le 2\\cdot 10^7}\\frac{(g(n+1)-S(n))\\log n}{\\sqrt n}.\n\\end{align*}\nThen for  ${2\\le n\\le 2\\cdot 10^7}$ we have\n$$\nS(n) \\le g(n)+c_1\\sqrt n,   \\qquad S(n) \\ge g(n+1)-c_2\\sqrt {n}.\n$$\nHence for ${2\\le x\\le 2\\cdot 10^7}$ we have \n\\begin{align*}\nS(x)&= S(\\lfloor x \\rfloor) \\le g(\\lfloor x \\rfloor)+c_1\\sqrt{\\lfloor x\\rfloor} \\le g(x)+c_1\\sqrt x,\\\\\nS(x)&= S(\\lfloor x \\rfloor) \\ge g(\\lfloor x \\rfloor+1)-c_2\\sqrt{\\lfloor x\\rfloor} \\ge g(x)-c_2\\sqrt x.\n\\end{align*}\nIn a similar way \nwe show that \nfor ${4\\cdot10^4\\le x\\le 2\\cdot 10^7}$ \n$$\ng(x)-c_4\\frac{\\sqrt x}{\\log x}\\le S(x) \\le g(x)+c_3\\frac{\\sqrt x}{\\log x}\n$$\nhaving used that $x\\mapsto \\sqrt{x}\/\\log x$ is increasing on\n$(e^2,\\infty)$.\nA computer-assisted calculation shows that \n$$\nc_1\\le 0.712, \\quad c_2\\le 1.010, \\quad\nc_3\\le 2.598, \\quad c_4\\le 2.267.\n$$\nWe verify this by means of a\n\\textsf{SAGE}~\\cite{sagemath} script\\footnote{A link to the  script\n  \\textsf{prop6\\_2.sage} is on  the second named\nauthor's homepage. The running time is roughly 30 minutes on a regular\ndesktop (Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM).} using the interval arithmetic \\textsf{MPFI} package~\\cite{MPFI}.\n\\end{proof}\n\n\\begin{corollary}\nLet~$A$ and~$B$ be positive real numbers satisfying \n$$\n0<A<B\\le 2\\cdot 10^7, \\qquad B\\ge 1.\n$$\nThen \n\\begin{equation}\n\\label{esqrtab}\n\\sum_{A<n\\le B}2^{\\omega(n)}\\le \\lambda_0 (B-A)(1+\\log B)+\\lambda_1(B-A)+1.722B^{1\/2}\n\\end{equation}\nIf, in addition to this, \n$\n{A\\ge 4\\cdot10^4}, \n$\nthen \n\\begin{equation}\n\\label{esqrtlogab}\n\\sum_{A<n\\le B}2^{\\omega(n)}\\le \\lambda_0 (B-A)(1+\\log B)+\\lambda_1(B-A)+4.865\\frac{B^{1\/2}}{\\log B}. \n\\end{equation}\n\\end{corollary}\n\n\n\n\\begin{proof}\nIn general we have\n\\begin{equation*}\nS(B)-S(A)=  \\sum_{A<n\\le B} 2^{\\omega(n)} = \\sum_{n =\n    \\lfloor{A}\\rfloor+1}^{\\lfloor{B}\\rfloor}2^{\\omega(n)} =\n  S(\\lfloor B\\rfloor) - S(\\lfloor A\\rfloor).\n\\end{equation*}\nNote that since $B>A>0$ we remark that\n\\begin{equation}\n\\label{eremark}\nB\\log B - A \\log A \\le (B-A)(1+\\log B).\n\\end{equation}\nWhen ${A\\ge 2}$ estimate~\\eqref{esqrtab} follows immediately\nfrom~\\eqref{esqrt} and~\\eqref{eremark}.\n\nLet us assume  that ${A<2}$, hence $\\lfloor A\\rfloor$ is $0$ or $1$\nand $S(\\lfloor A\\rfloor)=0$ or $1$, respectively. For $B\\ge 2$ we find\n$$\n\\sum_{A<n\\le B}2^{\\omega(n)} \\le \\lambda_0 B\\log\nB+\\lambda_1B+0.712B^{1\/2} -S(\\lfloor A\\rfloor),\n$$\nand one easily verifies that \n$$\n-S(\\lfloor A\\rfloor)+\\lambda_0A(1+\\log B) +\\lambda_1A - \\lambda_0\n\\le 1.010B^{1\/2}\n$$\nby considering the cases $A\\in (0,1)$ and $A\\in [1,2)$ separately.\n  This implies (\\ref{esqrtab}). \nAnd if ${B<2}$, then \n${\\sum_{A<n\\le B} 2^{\\omega(n)}\\le 1\\le 1.722\nB^{1\/2}}$.  \nAs $B\\ge 1$, again we obtain (\\ref{esqrtab}). \n\nFinally, for ${4\\cdot10^4\\le A<B\\le 2\\cdot 10^7}$ estimate~\\eqref{esqrtlogab}\nfollows\nfrom~\\eqref{esqrtlog} and~\\eqref{eremark}.  \n\\end{proof}\n\n\n\n\\subsection{Bounding ${\\mathcal{C}}_\\varepsilon(\\Delta)$ for ${10^{10}\\le|\\Delta|< 10^{15}}$}\n\\label{ssnewceps}\n\nNow we can obtain a cardinal refinement of Theorem~\\ref{thceps}  for discriminants in the range ${10^{10}\\le |\\Delta|< 10^{15}}$.  We need a technical lemma using our\nnotation~\\eqref{eq:standardnotation}. \n\n\\begin{lemma}\n  \\label{lem:varsigma01}\n  Let $n$ be an integer with $1\\le n\\le 3.2\\cdot 10^7$.\n  \\begin{enumerate}[label={(\\roman*)}]\n  \\item \n  \\label{isigone} We have\n    $\\sigma_1(n)\/n \\le \\sigma_1(21621600)\/21621600= 3472\/715$. \n  \\item \n  \\label{isigzero} We have\n    $\\sigma_0(n)\\le 8.5 n^{1\/4}$. \n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThis can be proved in at least two ways. In our relatively small range, we can\nperform a quick computer-assisted calculation\\footnote{A link to the \\textsf{PARI} script \\textsf{lemma6\\_4.gp} is on the second-named\nauthor's homepage. The running time is under 2 minutes on a regular\ndesktop (Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM).}.  Alternatively, one can use the\n\\textit{On-Line Encyclopedia of Integer Sequences}~\\cite{oeis}. Inspecting T. D. Noe and D. Kilminster's\ntable in entry \\textsf{A004394}, one can deduce~\\ref{isigone}. Similarly,~\\ref{isigzero} follows from inspection of T. D. Noe's table in \\textsf{A002182}.\n\\end{proof}\n\n\n\\begin{proposition}\n  \\label{prop:Cepsmidrange}\nIn the set-up of Theorem~\\ref{thceps} assume that \n\\begin{equation}\n\\label{emidrange}\n{10^{10} \\le |\\Delta|< 10^{15}}.\n\\end{equation}\nThen\n\\begin{align*}\n{\\mathcal{C}}_\\varepsilon(\\Delta) &\\le (8\\varepsilon^2 + 0.811\\varepsilon)|\\Delta|^{1\/2} \\log|\\Delta|\n+(28\\varepsilon^2 +2.829\\varepsilon)|\\Delta|^{1\/2}  \\\\\n&\\hphantom{\\le}+89\\varepsilon {|\\Delta|^{3\/8}}+\n31.06\\frac{|\\Delta|^{1\/4}}{\\log|\\Delta|}\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof}\nAs in the proof of Theorem~\\ref{thceps}, we\nset\n$$\nI=\\left(\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\frac{|\\Delta|^{1\/2}}{\\sqrt3}\\right]. \n$$\nWe again want to count pairs of integers $(a,b)$ such that \n\\begin{align*}\na\\in I,\\qquad b\\in [-a,-a(1-2\\varepsilon))\\cup (a(1-2\\varepsilon),a], \\qquad b^2\\equiv \\Delta\\bmod a. \n\\end{align*}\nLemmas~\\ref{lrootsmoda} and~\\ref{ltrivi} \nimply that,  for\nevery fixed~$a$, the number of  suitable~$b$ \ndoes not exceed ${(4\\varepsilon\\gcd_2(a,\\Delta)+2) 2^{\\omega(a\/\\gcd(a,\\Delta))+1}}$.\nHence\n\\begin{equation}\n\\label{ethreesums}\n{\\mathcal{C}}_\\varepsilon(\\Delta) \\le 8\\varepsilon\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta)\n2^{\\omega(a\/\\gcd_2(a,\\Delta)^2)}\n+ 4\\sum_{a\\in I\\cap\\Z}2^{\\omega(a)\n\\end{equation}\nwhere we used that $\\gcd_2(a,\\Delta)^2$ divides $\\gcd(a,\\Delta)$. \nWe estimate each of the terms separately.\n\n\n\\subsubsection*{Estimating the first term in~\\eqref{ethreesums}}\nRecall that $\\Delta \/ {\\tilde f}^2$ is a square-free integer. We have \n\\begin{equation}\n\\label{elefthand}\n\\sum_{a\\in I\\cap\\Z} {\\gcd}_2(a,\\Delta) 2^{\\omega(a\/\\gcd_2(a,\\Delta)^2)} \\le \\sum_{d\\mid {\\tilde f}}d\\sum_{a\\in I\\cap d^2\\Z}  2^{\\omega(a\/d^2)}.\n\\end{equation}\nTo estimate the inner sum, write ${a\\in I\\cap d^2\\Z}$ as ${a=d^2a'}$ with \n$$\na'\\in d^{-2}I = \\left(d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3}\\right].\n$$\nWe estimate $\\sum_{a'\\in d^{-2}I\\cap\\Z} 2^{\\omega(a')}$  using~\\eqref{esqrtab} with\n$$\nA=d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\quad B=d^{-2}\\frac{|\\Delta|^{1\/2}}{\\sqrt3}. \n$$ \nSince ${|\\Delta|\\le  10^{15}}$, we have ${B\\le 2\\cdot 10^7}$. From \n$$\nB-A\\le \\frac23 d^{-2}|\\Delta|^{1\/2}\\varepsilon, \\quad \\log B \\le \\frac12\\log |\\Delta|-\\frac12\\log 3,\n$$\nwe obtain \n$$\n\\sum_{a'\\in d^{-2}I\\cap \\Z}  2^{\\omega(a')}\\le \\frac{|\\Delta|^{1\/2}}{3d^2}(\\lambda_0 \\log |\\Delta| +2\\lambda_0+2\\lambda_1-\\lambda_0\\log3)\\varepsilon+\\frac{1.722|\\Delta|^{1\/4}}{\\sqrt[4]{3}d}  \n$$\nas long as $B\\ge 1$. If $B<1$, then the sum on the left is $0$ and the\ninequality remains valid as the right side is clearly positive. \nHence the left-hand side of~\\eqref{elefthand} is bounded by \n\\begin{align*}\n\\frac{|\\Delta|^{1\/2}}{3}(\\lambda_0 \\log |\\Delta|+ 2\\lambda_0+2\\lambda_1-\\lambda_0\\log3)\\varepsilon\\frac{\\sigma_1({\\tilde f})}{{\\tilde f}\n+\\frac{1.722|\\Delta|^{1\/4}}{\\sqrt[4]{3}}\\sigma_0({\\tilde f})\n\\end{align*}\nwhere we use  notation~\\eqref{eq:standardnotation} and the identity\n$\\sigma_1({\\tilde f})\/{\\tilde f} = \\sum_{d \\mid {\\tilde f}} d^{-1}$.\n\nRecall that ${\\tilde f} \\le |\\Delta|^{1\/2}\\le 3.2\\cdot 10^7$.\nHence Lemma \\ref{lem:varsigma01} implies that\n$$\n\\sigma_1({\\tilde f})\/{\\tilde f} \\le\n3472\/715, \\qquad \\sigma_0({\\tilde f})\\le 8.5 {\\tilde f}^{1\/4} \\le 8.5|\\Delta|^{1\/8}. \n$$\nTaking into account the factor $8\\varepsilon$, the first term in~\\eqref{ethreesums} is thus at most\n\\begin{alignat*}1\n   |\\Delta|^{1\/2} &(8\\log|\\Delta| + 28)\\varepsilon^2 +\n   89{|\\Delta|^{3\/8}}\\varepsilon.\n\\end{alignat*}\n\n\\subsubsection*{Estimating the second  term in~\\eqref{ethreesums}}\nSet\n$$\nA=\\frac{|\\Delta|^{1\/2}}{\\sqrt3+2\\varepsilon}, \\qquad B=\\frac{|\\Delta|^{1\/2}}{\\sqrt3}.\n$$ \nFrom~\\eqref{emidrange} and ${0<\\varepsilon \\le 1\/3}$ we deduce ${A\\ge 4\\cdot10^4}$  and ${B\\le\n2\\cdot 10^7}$. This allows us to apply~\\eqref{esqrtlogab}, and we obtain\n\\begin{align*}\n\\sum_{a\\in I\\cap\\Z}2^{\\omega(a)} &\\le \\frac{|\\Delta|^{1\/2}}{3}(\\lambda_0 \\log |\\Delta|+2\\lambda_0+2\\lambda_1 -\\lambda_0\\log3)\\varepsilon\\\\\n&\\hphantom{\\le}+\\frac{4.865|\\Delta|^{1\/4}}{\\sqrt[4]{3}\\log(|\\Delta\/3|^{1\/2})}\\\\\n &\\le \\frac{|\\Delta|^{1\/2}}{3}(\\lambda_0 \\log\n|\\Delta|+2\\lambda_0+2\\lambda_1 -\\lambda_0\\log3)\\varepsilon\\\\\n&\\hphantom{\\le}+\\frac{2\\cdot\n  4.865|\\Delta|^{1\/4}}{\\sqrt[4]{3}\\log |\\Delta|}\n\\frac{\\log10^{10}}{\\log(10^{10}\/3)}.\n\\end{align*}\nwhere  for the last estimate we used the assumption\n ${|\\Delta|\\ge\n  10^{10}}$. \nMinding the factor~$4$ we find that \nthe second term in~\\eqref{ethreesums} is at most\n\\begin{equation*}\n  |\\Delta|^{1\/2} (0.811 \\log|\\Delta| + 2.829 )\\varepsilon  + 31.06\n   \\frac{|\\Delta|^{1\/4}}{\\log |\\Delta|}. \n\\end{equation*}\nThis concludes our proof of Proposition \\ref{prop:Cepsmidrange}.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thmidrange}}\n\\label{ssproofmidrange}\nSuppose that~$\\alpha$ is a singular unit of discriminant~$\\Delta$\nand set ${X=|\\Delta|}$. Assuming  that ${10^{10}\\le X< 10^{15}}$,\nwe arrive at a contradiction. \nAs in   Section~\\ref{stenfourteen}  we use the estimates~\\eqref{etrivx} and~\\eqref{ecolmx} which follow from Propositions~\\ref{ptriv}\nand~\\ref{phlb2}, respectively. \n\nWe may no longer use Corollary~\\ref{cupper}, because its hypothesis ${X\\ge 10^{14}}$ is not valid in our current range. Instead, we will apply Theorem~\\ref{theceps}   directly now. For this, let\n${\\varepsilon \\in (0,4\\cdot 10^{-3}]}$. \n\n{\\sloppy\n\nWe define~$Y$ as in (\\ref{ey}) and recall that ${Y\\le 0.01 + {\\mathrm{h}}(\\alpha)}$. We find\n\\begin{equation}\n  \\label{eq:midrangeYbound}\n  Y \\le 3 \\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta)}\n  \\log X + 3 \\log(\\varepsilon^{-1}) - 10.65. \n\\end{equation}\nUsing Proposition \\ref{prop:Cepsmidrange} we find\n\\begin{alignat*}1\n  1&\\le 3 (8\\varepsilon^2 + 0.811\\varepsilon)\n  \\frac{X^{1\/2}(\\log X)^2}{Y{\\mathcal{C}}(\\Delta)} + 3 {(28\\varepsilon^2 + 2.829\n    \\varepsilon)}\\frac{X^{1\/2}\\log X}{Y{\\mathcal{C}}(\\Delta)}  \\\\&\n\\hphantom{\\le}    +\n3 \\frac{89 \\varepsilon X^{3\/8}\\log X}{Y{\\mathcal{C}}(\\Delta)}  \n + 3\\cdot 31.06\n  \\frac{X^{1\/4}}{Y{\\mathcal{C}}(\\Delta)} \n+ \\frac{3\\log(\\varepsilon^{-1}) - 10.65}{Y}.\n\\end{alignat*}\nFor all but the final term on the right we use ${Y{\\mathcal{C}}(\\Delta)\\ge \\pi X^{1\/2}}$\nand for the remaining term we use\n${Y\\ge \\frac{3}{\\sqrt{5}} \\log X - 9.78 > 0}$, as ${X\\ge  10^{5}}$,  \nto get\n\\begin{equation}\n  \\label{eq:1lowerbound}\\begin{aligned}\n  1&\\le \\frac{3}{\\pi}(8\\varepsilon^2 + 0.811\\varepsilon) (\\log X)^2\n  +\\frac{3}{\\pi}(28\\varepsilon^2 + 2.829\\varepsilon) \\log X\\\\\n  &\\hphantom{\\le}+  \\frac{267}{\\pi}\\varepsilon\\frac{\\log X}{X^{1\/8}} \n  +\\frac{93.18}{\\pi X^{1\/4}}\n   +\n    \\frac{3\\log(\\varepsilon^{-1})-10.65}{\\frac{3}{\\sqrt 5}\\log X - 9.78}.\n\\end{aligned}\n\\end{equation}\nOur choice is ${\\varepsilon = \n10^{-4}}$.\nThe first two terms in the right-hand side of~\\eqref{eq:1lowerbound} are \nmonotonously increasing, and the remaining three terms are  decreasing for ${X\\in[10^{10}, 10^{15})}$; note that ${x\\mapsto(\\log x)\/x^{1\/8}}$ is\ndecreasing for ${x \\ge 3000 > e^8}$. Using ${X < 10^{15}}$  for the first two terms and ${X\\ge  2\\cdot 10^{10}}$ for the\n remaining three terms, we see that the right-hand side of~\\eqref{eq:1lowerbound} is strictly smaller than $0.962$ if\n${X \\in [2\\cdot10^{10}, 10^{15})}$. Similarly, we infer that it is strictly smaller than $0.960$ if ${X \\in [10^{10}, 2\\cdot10^{10})}$. This completes the proof of Theorem~\\ref{thmidrange}. \n\\qed\n\n}\n\n\\section{Handling the low-range \\texorpdfstring{$3\\cdot 10^{5}\\le |\\Delta|< 10^{10}$}{3.10to5<|Delta|<10to10}}\n\\label{slowrange}\nWe now deal with the low-range ${|\\Delta|\\in [3\\cdot10^5, 10^{10})}$. For this range the upper\nbound on ${\\mathcal{C}}_\\varepsilon(\\Delta)$ arises from a computer-assisted search algorithm.\n\n \n\nWe prove the following. \n\\begin{theorem}\n  \\label{thlowrange}\n  Let~$\\Delta$  be the discriminant of a singular unit. Then\n ${|\\Delta|\\notin [3\\cdot10^5, 10^{10})}$. \n\\end{theorem}\n\nThe proof relies on the following lemma. \n\n\\begin{lemma}\n  \\label{lem:lowrangeCebound}\n  Let $\\Delta$ be the discriminant of a singular modulus.\n  \\begin{enumerate}[label={(\\roman*)}]\n  \\item \n  \\label{itenten} If ${10^7\\le|\\Delta|<  10^{10}}$ and ${\\varepsilon = 10^{-3}}$, then\n    ${{\\mathcal{C}}_\\varepsilon(\\Delta)\\le 16}$.\n      \\item \n      \\label{itenseven}\n      If ${3\\cdot10^5\\le|\\Delta|< 10^7}$ and ${\\varepsilon = 4\\cdot 10^{-3}}$, then\n    ${{\\mathcal{C}}_\\varepsilon(\\Delta)\\le 6}$. \n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $X_{\\min}$ and $X_{\\max}$ be positive integers satisfying ${ X_{\\min}<X_{\\max}}$, and let ${\\varepsilon \\in (0, 1\/3]}$. We want to bound ${{\\mathcal{C}}_\\varepsilon(\\Delta)}$ for all~$\\Delta$ in the interval ${[-X_{\\max},-X_{\\min}]}$. \n \nRecall that ${\\mathcal{C}}_\\varepsilon(\\Delta)$ counts the triples    $(a,b,c)$ satisfying~\\eqref{ekuh} such that \n$$\n\\tau=\\tau(a,b,c)= \\frac{b + \\sqrt{\\Delta}}{2a}\n$$\nsatisfies\n  \\begin{equation}\n    \\label{eq:zrestriction}\n    \\min\\{|\\tau-\\zeta_3|,|\\tau-\\zeta_6|\\} < \\varepsilon. \n  \\end{equation}\nLemmas~\\ref{lgauss}\\ref{iadelta} and~\\ref{lshort} imply that such triples satisfy \n\\begin{align*}\n\\frac{|\\Delta|^{1\/2}}2 & \\le \\hphantom{|}c \\hphantom{|}\\le \\frac{|\\Delta|^{1\/2}}{\\sqrt3}(1+\\sqrt3\\varepsilon+\\varepsilon^2),\\\\\nc\/(1+\\sqrt3\\varepsilon+\\varepsilon^2)&<\\hphantom{|}a\\hphantom{|}\\le c,\\\\\na(1-2\\varepsilon)&<|b|\\le a. \n\\end{align*}\nNote that, since ${\\varepsilon \\in (0, 1\/3]}$, we have ${b\\ne 0}$ and \n${(1+\\sqrt3\\varepsilon+\\varepsilon^2)\/\\sqrt3<1}$. \n\nHence, to bound \n${\\mathcal{C}}_\\varepsilon(\\Delta)$ on the interval ${[-X_{\\max},-X_{\\min}]}$,\nit suffices, for every~$\\Delta$ in this interval, to count the triples $(a,b,c)$ satisfying \n\\begin{align*}\n{X_{\\min}^{1\/2}}\/2 & \\le c \\le X_{\\max}^{1\/2},\\\\\nc\/(1+\\sqrt3\\varepsilon+\\varepsilon^2)&<a\\le c,\\\\\na(1-2\\varepsilon)&<b\\le a \n\\end{align*}\nand ${b^2-4ac=\\Delta}$, and multiply the maximal count by~$2$ (because we counted only triples with positive~$b$). \nWe phrase this counting procedure formally as   Algorithm~\\ref{algo:CCDelta}. \n\n\nFor a correct implementation, we have to avoid floating point arithmetic in determining\nthe upper bounds on~$a$ and~$b$ used in the inner two for-loops. For this, we note that ${\\varepsilon = 10^{-3}}$\nimplies that \n$$\n0.998 c \\le c\/(1+\\sqrt3\\varepsilon+\\varepsilon^2), \\qquad 0.998a=a(1-2\\varepsilon). \n$$\nSimilarly, ${\\varepsilon = 4\\cdot 10^{-3}}$\nimplies that \n$$\n0.993 c \\le c\/(1+\\sqrt3\\varepsilon+\\varepsilon^2), \\qquad 0.992a=a(1-2\\varepsilon). \n$$\nAs we are only interested in an upper bound on ${\\mathcal{C}}_\\varepsilon(\\Delta)$ for these two specific values of~$\\varepsilon$, we use these\nweaker rational bounds in our implementation of\nAlgorithm~\\ref{algo:CCDelta}  by means of a\n\\textsf{C}-program\\footnote{A link to our program\n  \\textsf{algorithm1.c} is on the\nsecond-named author's homepage. The running time on a regular desktop\n(Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM) was under a minute for item~\\ref{itenten} and a few milliseconds for item~\\ref{itenseven}. Its memory\nusage for ${[X_{\\min},X_{\\max}] = [1, 10^{10}]}$ is significant (5 GB) but this can be overcome by splitting\n${[1, 10^{10}]}$ into subintervals and running the program separately for each interval. This feature\nis also implemented in our program through the macro \\textsf{DISC\\underline{\\ }BLOCK\\underline{\\ }SIZE}.}. It\nverifies directly the assertions of the lemma.\n\\end{proof}\n\n\n\n    \\begin{algorithm}[t]{\\footnotesize\n      \\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n      \n    \\Input{Two positive integers $X_{\\mathrm{min}} <\n      X_{\\mathrm{max}}$ and $\\varepsilon > 0$}\n    \n    \n    \\Output{an upper bound for ${\\mathcal{C}}_\\varepsilon(\\Delta)$ for all\n      discriminants $\\Delta\\in [-X_{\\mathrm{max}},-X_{\\mathrm{min}}]$}\n    \\BlankLine\n\n    $counter \\leftarrow$ pointer to array of length\n    $X_{\\mathrm{max}}-X_{\\mathrm{min}}+1$ initialized to $0$\\;\n\n    $bound \\leftarrow 0$\\;\n    \n    \\For{$c\\leftarrow \\lfloor X_{\\mathrm{min}}^{1\/2}\/2\\rfloor$ \\KwTo\n      $\\lfloor X_{\\mathrm{max}}^{1\/2}\\rfloor$}{\n      \\For{$a\\leftarrow \\lfloor c\/(1+\\sqrt 3 \\varepsilon+\\varepsilon^2) \\rfloor$ to $c$}{\n        \\For{$b\\leftarrow \\lfloor (1-2\\varepsilon)a\\rfloor$ to $a$}{\n          $X\\leftarrow 4ac-b^2$\\;\n          \\If{$X\\ge X_{\\mathrm{min}}$  and $X \\le X_{\\mathrm{max}}$\n          }{$pos \\leftarrow X-X_{\\mathrm{min}}$\\;\n            $counter[pos] \\leftarrow counter[pos]+2$\\;\n            \\lIf{$counter[pos] > bound$}{$bound\\leftarrow counter[pos]$}\n            }\n        }\n    }}\n    \\Return{$bound$}\\;\n    \\caption{Compute an upper bound for ${\\mathcal{C}}_{\\epsilon}(\\Delta)$ in the\n      range ${\\Delta\\in [-X_{\\mathrm{max}},-X_{\\mathrm{min}}]}$}\\label{algo:CCDelta}\n      }%\n    \\end{algorithm}\n    \n\n\n\\begin{proof}[Proof of Theorem~\\ref{thlowrange}]\n  Assume that~$\\alpha$ is a singular unit of discriminant ${\\Delta\\in (-10^{-10}, -3\\cdot10^5]}$.\nLet ${0<\\varepsilon\\le 4\\cdot 10^{-3}}$. We set again~$Y$  as in (\\ref{ey}).\n  As in the proof of Theorem~\\ref{thmidrange} we find (\\ref{eq:midrangeYbound}).  \n \n \nWe infer that\n  \\begin{alignat*}1\n    1 &\\le 3\\frac{{\\mathcal{C}}_\\varepsilon(\\Delta)}{{\\mathcal{C}}(\\Delta) Y} \\log X +\n    \\frac{3\\log(\\varepsilon^{-1}) - 10.65}{Y}\\\\\n    & \\le \\frac {3{\\mathcal{C}}_\\varepsilon(\\Delta)}{\\pi} \\frac{\\log X}{X^{1\/2}} +\n    \\frac{3\\log(\\varepsilon^{-1}) - 10.65}{\\frac{3}{\\sqrt{5}}\\log X - 9.78}\n  \\end{alignat*}\nwhere we use ${{\\mathcal{C}}(\\Delta)Y \\ge \\pi X^{1\/2}}$\n  and ${Y\\ge \\frac{3}{\\sqrt{5}} \\log X - 9.78}$. \n\n  If ${X\\in [10^7,10^{10})}$ then we set ${\\varepsilon = 10^{-3}}$ and use the estimate\n  ${{\\mathcal{C}}_{\\varepsilon}(\\Delta) \\le 16}$ from Lemma~\\ref{lem:lowrangeCebound}\\ref{itenten}. \n  Recall that ${x\\mapsto (\\log x)\/x^{1\/2}}$\n  is decreasing for ${x\\ge e^2}$. So we find\n  \\begin{equation*}\n    1 \\le \\frac{3\\cdot 16}{\\pi}\\frac{\\log(10^7)}{10^{7\/2}} +\n    \\frac{3\\log(1000) - 10.65}{\\frac{3}{\\sqrt{5}} \\log(10^7)- 9.78}< 0.929,\n  \\end{equation*}\n  a contradiction.\n\nWhen ${X\\in [3\\cdot10^5,10^7)}$  we set ${\\varepsilon = 4\\cdot10^3}$.  Then\n  ${{\\mathcal{C}}_{\\varepsilon}(\\Delta)\\le 6}$ by  Lem\\-ma~\\ref{lem:lowrangeCebound}\\ref{itenseven}.\nUsing \n   ${X\\ge 3\\cdot 10^5}$ we find as before \n  \\begin{equation*}\n    1 \\le \\frac{3\\cdot 6}{\\pi}\\frac{\\log(3\\cdot 10^5)}{(3\\cdot 10^{5})^{1\/2}} +\n    \\frac{3\\log(250) - 10.65}{\\frac{3}{\\sqrt{5}} \\log(3\\cdot 10^{5})- 9.78}< 0.961,\n  \\end{equation*}\n  another contradiction which completes this proof. \n\\end{proof}\n\n\n\\section{The extra low-range}\n\n\\label{sfinal}\n\nThe results of the three previous sections reduce the proof of Theorem~\\ref{thmain}  to the\nfollowing assertion.\n\n\n\\begin{theorem}\n\\label{thm:habeggerspari}\n    Let~$\\Delta$ be the discriminant of a singular unit. Then \n  ${|\\Delta|\\ge 3\\cdot 10^5}$. \n\\end{theorem}\n\\begin{proof}\n  Let $\\alpha$ be a singular unit of discriminant $\\Delta$.  We write\n  ${X=|\\Delta|}$. \nWe may assume that ${X\\ge 4}$ because the only singular modulus of discriminant $-3$ is\n${j(\\zeta_3)=0}$, \nwhich is  not an algebraic unit.\n\n\nRecall from Section~\\ref{sceps} that the Galois conjugates of~$\\alpha$ are precisely the singular moduli\n$j(\\tau)$, where ${\\tau = \\tau(a,b,c)}$  with $(a,b,c)$ as in~\\eqref{ekuh}. The imaginary part of  such~$\\tau$ is\n$X^{1\/2}\/(2a)$ and ${a\\le (X\/3)^{1\/2}}$\nby Lemma~\\ref{lgauss}\\ref{iadelta}. \n  Lemma~\\ref{lttsn}  implies that\n$$\n|j(\\tau)|\\ge e^{2\\pi X^{1\/2}\/(2a)} -2079\n  =e^{\\pi X^{1\/2}\/a}-2079 > 23^{X^{1\/2}\/a}-2079\n$$\n  as ${e^\\pi > 23}$. Using Lemmas~\\ref{lanal} and~\\ref{lliouv}, we find that\n$$\n|j(\\tau)| \\ge 42700 \\min \\left\\{ \\frac{\\sqrt3}{4X},4\\cdot 10^{-3}\\right\\}^3.\n$$\nThese\n  bounds together show that\n  \\begin{equation}\n\\label{eq:finaljlb}\n    |j(\\tau)|\\ge \\max\\left\\{ 23^{\\lfloor X^{1\/2}\/a\\rfloor}-2079,\n    42700\\min\\Bigl\\{\\frac2{5X},\\frac1{250}\\Bigr\\}^3\\right\\}. \n  \\end{equation}\nBased on this observation,   Algorithm~\\ref{algo:exclude} prints a list of discriminants of potential singular\nunits in the range ${[-X_{\\max},-4]}$. For this purpose, it computes a \nrational lower bound~$P$ for the absolute value of the\n$\\Q(\\alpha)\/\\Q$-norm  of each singular moduli in this range. Those singular\nmoduli where ${P \\le 1}$ are then flagged as potential singular units.\n\n We have implemented this algorithm as a \\textsf{PARI}\n script\\footnote{A link to our \\textsf{PARI} script  \\textsf{algorithm2.gp}\nis on  the second-named author's homepage. The running time is about 23 minutes  on a regular desktop computer (Intel Xeon CPU E5-1620 v3, 3.50GHz, 32GB RAM). The only floating point operation used  approximates $X^{1\/2}$\n  which leads to  ${n=\\lfloor X^{1\/2}\/a\\rfloor}$.\nTo rule out a rounding error in the floating point arithmetic we\ncompare $(an)^2$  with~$X$ in our implementation.}. \nThe script flags only $-4$, $-7$ and $-8$ as discriminants of potential singular units. The\nsingular moduli of these discriminants are well-known \\cite[(12.20)]{Cox}: they are $12^3$, $-15^3$ and $20^3$, respectively. None of them is a unit, which concludes the proof. \n\\end{proof} \n  \n  \\begin{algorithm}[t]{\\footnotesize\n          \\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n          \\Input{An integer $X_{\\mathrm{max}}\\ge 1$}\n          \\Output{Print a list containing all discriminants\n            in $[-X_{\\mathrm{max}},-4]$  that are attached to a potential\n            singular unit.}\n\n          \\For{$X\\leftarrow 4$ \\KwTo $X_{\\mathrm{max}}$}{\n              $\\Delta \\leftarrow -X$\\;\n            \n            \\lIf{$\\Delta\\equiv 2 \\text{ or }3 \\mod 4$}{\n              next $X$\n             }\n            \n\n\n$P\\leftarrow 1$\\;\n            \\For{$a\\leftarrow 1$ \\KwTo $\\lfloor \\sqrt{X\/3}\\rfloor$} {\n\n              $n\\leftarrow \\lfloor X^{1\/2}\/a\\rfloor$\\;\n              \\For{$b\\leftarrow -a+1$ \\KwTo $a$}{\n                \\lIf{$b^2  \\not\\equiv \\Delta \\mod 4a$}{\n                  next $b$\n                }\n                $c\\leftarrow (b^2-\\Delta)\/(4a)$\\;\n                \\lIf{$a>c$}{next $b$}\n                \\lIf{$a=c \\text{ and } b<0$}{next $b$}\n                \\lIf{$\\mathrm{gcd}(a,b,c)\\not=1$}{next $b$}\n               \n               \n               \n               \n  $P\\leftarrow P \\cdot \\max\\{23^{n\n        }-2079,42700\\min\\{2\/(5X),1\/250\\}^3\\}$\\;\n}}\n\n\\lIf{$P\\le 1$}{print $\\Delta$}}\n\\caption{Exclude singular units}          \\label{algo:exclude}\n}%\n  \\end{algorithm}\n \n\n \n\n\nAs indicated in the introduction, Theorem~\\ref{thmain} is the combination of Theorems~\\ref{thhighrange},~\\ref{thmidrange},~\\ref{thlowrange} and~\\ref{thm:habeggerspari}. \n\n\n{\\footnotesize\n\n\\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\ \\ \\ \\ The time evolution of coherent states (CS) has attracted a great\ndeal of attention since the introduction of Glauber's CS of the harmonic\noscillator {\\normalsize \\cite{Glauber63}}. Of\nparticular interest has been the determination of the Hamiltonian operator\nfor which an initial coherent state remains coherent under time evolution.\nIt is established that this Hamiltonian has the form of the nonstationary\nbosonic forced oscillator Hamiltonian {\\normalsize \\cite{Glauber66, Mehta66,\n Stoler75, Kano76}:  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\n\\begin{equation}\nH_{\\mathrm{cs}}=\\omega (t)a^{\\dagger }a+f(t)a^{\\dagger }+f^{\\ast }(t)a+\\beta\n(t),  \\label{Hcs}\n\\end{equation}%\nwhere $\\omega (t)$ and $\\beta (t)$ are arbitrary real functions of time $t$,\nand $f(t)$ is arbitrary complex function. \n$H_{\\mathrm{cs}}$ is a particular case of the nonstationary forced oscillator\nHamiltonian for which the exact time evolution of CS has been obtained in\n\\cite{MMT70,Holz70} by first constructing boson ladder operator dynamical %\ninvariants according to the Lewis and Riesenfeld scheme of time\ndependent invariants \\cite{Lewis69}.\n\nOur purpose in the present article is to study the dynamical invariants and\ntime evolution of CS for the {\\it fermionic} forced oscillator (FFO), which %\nin fact is the general (one mode) Hamiltonian.\n\nThe organization of the article is as follows. In Sec. 2 we construct fermionic %\nladder operator dynamical invariants and the corresponding Lewis-Riesenfeld\nHermitian invariant {\\normalsize \\cite{Lewis69}}, following the scheme related %\nto the boson system {\\normalsize \\cite{MMT70}}. Using these invariants, we\nconstruct in Sec. 3 fermionic CS and Fock states of FFO system as eigenstates %\nof the constructed invariant fermionic annihilation operator $B(t)$ and %\n$B^\\dagger(t)B(t)$ correspondingly. These CS can\nrepresent (under appropriate initial conditions) the exact time-evolution of\ninitial canonical fermionic CS. Finally the relation of the invariant ladder %\noperators method {\\normalsize \\cite{MMT70, Holz70}} to the Lewis-Riesenfeld method\n{\\normalsize \\cite{Lewis69}} is briefly described on the example of FFO. The\npaper ends with concluding remarks.\\\n\n\\medskip\n\n\\section{FFO and invariant ladder operators}\n\nWe consider the single nonstationary fermionic forced oscillator (FFO)\ndescribed by the following Hamiltonian,%\n\\begin{equation}\nH_{\\!f}=\\omega (t)b^{\\dagger }b+f(t)b^{\\dagger }+f^{\\ast }(t)b+g(t),\n\\label{Hf}\n\\end{equation}%\nwhere $\\omega (t)$ and $g(t)$ are arbitrary real functions of time, $f(t)$\nis arbitrary complex function, $b$ and $b^{\\dagger }$ are fermion\nannihilation and creation operators respectively, which obey to the fermion\nalgebra:\n\\begin{equation}\n\\{b,b^{\\dagger }\\} = 1,\\text{ \\ }%\nb^{2}=b^{\\dagger }{}^{2}=0,  \\label{013}\n\\end{equation}%\nwhere $\\{b,b^{\\dagger }\\}\\equiv bb^{\\dagger }+b^{\\dagger }b$.\nDue to the nilpotency of the fermionic operators $b$, $b^\\dagger$\nthe operator $H_{\\!f}$ represents the most general (one mode) fermionic %\nHamiltonian.\n\nThe Hilbert space $\\mathcal{H}$ of the single-fermion system is spanned by\nthe two eigenstates $\\left\\{ \\left\\vert 0\\right\\rangle ,\\left\\vert\n1\\right\\rangle \\right\\} $ of  number operator $b^{\\dagger }b$:\\, %\n$b^{\\dagger }b\\left\\vert n\\right\\rangle =n\\left\\vert n\\right\\rangle ,\\text{ \\ }n=0,1$.\nThe operators $b$ and $b^{\\dagger }$\\ allow transitions between number states,\n\\begin{equation}\nb\\left\\vert 0\\right\\rangle =0,\\text{\\ }b\\left\\vert 1\\right\\rangle\n=\\left\\vert 0\\right\\rangle \\,,\\text{ \\ }b^{\\dagger }\\left\\vert\n1\\right\\rangle =0,\\text{ }b^{\\dagger }|0\\rangle =|1\\rangle .\\,\n\\end{equation}%\nThe form of the Hamiltonian ({\\normalsize \\ref{Hf}}) is a Hermitian linear\ncombination of $b$, $b^{\\dagger }$ and $N = b^{\\dagger }b$. The fermion number\noperator $N$ obey the relation $N^{2}=N$ and the three operators $b$, $%\nb^{\\dagger }$and $N$ close under commutation the algebra:\n\\begin{equation}\n\\left[ b,N\\right] =b,\\text{ }\\left[ b^{\\dagger },N\\right] =-b^{\\dagger },%\n\\text{ }\\left[ b,b^{\\dagger }\\right] =1-2N,\\text{ \\ \\ \\ }\n\\end{equation}%\nLet us note that linear combinations of $b^{\\dagger }$, $b$ and $N$ produce\nthe half-spin operators $J_{i}$,\n\\begin{equation}\nJ_{1}=\\tfrac{1}{2}(b^{\\dagger }+b),\\quad J_{2}=\\tfrac{1}{2\\ri}(b^{\\dagger\n}-b),\\quad J_{3}=b^{\\dagger }b-\\tfrac{1}{2},  \\label{30a}\n\\end{equation}%\nclosing the \\textit{su}(2) algebra: $\\left[ J_{k},J_{l}\\right] =i\\epsilon\n_{klm}J_{m}.$\n\nIt is convenient to use raising and lowering operators $J_{\\pm }=J_{1}\\pm\niJ_{2}$ which satisfy the following commutation relation: $\\left[ J_{+},J_{-}%\n\\right] =2J_{3},\\text{ }\\left[ J_{3},J_{\\pm }\\right] =\\pm J_{\\pm }\\text{\\ }$%\n, where $J_{+}=b^{\\dagger },$ $J_{-}=b.$ So that in terms of these half spin\noperators the Hamiltonian ({\\normalsize \\ref{Hf}}) takes the form\n\\begin{equation}\nH_{\\!f}=\\omega (t)J_{3}+f(t)J_{+}+f^{\\ast }(t)J_{-}+g(t)+\\tfrac{\\omega (t)}{2%\n}.  \\label{Hf 2}\n\\end{equation}%\n\nOur task is the construction of the time-dependent invariants for the system\n{\\normalsize (\\ref{Hf})}, {\\normalsize (\\ref{Hf 2})}. The defining equation\nof the invariant operator $B(t)$ for a quantum system with Hamiltonian $H(t)$\nis%\n\\begin{equation}\n\\frac{\\partial }{\\partial t}B(t)-i[B(t),H]=0  \\label{19}\n\\end{equation}\nFormal solutions to Eq. {\\normalsize (\\ref{19})} are operators $%\nB(t)=U(t)B(0)U^{\\dagger }(t)$, where $U(t)$ is the evolution operator of the\nsystem, $U =T\\exp [-\\ri\\tint_{0}^{t}H(t^{^{\\prime }}){\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t^{^{\\prime }}]$. \nIn our case of FFO {\\normalsize (\\ref{Hf})}, {\\normalsize (\\ref{Hf 2})} we look\nfor the non-Hermitian invariants $B(t)$, $B^\\dagger(t)$ of the form of linear %\ncombination of the $SU(2)$ generators {\\normalsize (\\ref{30a})}, \\\n\\begin{equation}\n\\begin{aligned} B = \\nu_{-}(t)J_{-} + \\nu_{+}(t) J_{+} + \\nu_3 (t) J_{3}\n,\\\\ B^\\dagger = \\nu_{-}^* (t)J_{+} + \\nu_{+}^*(t) J_{-} + \\nu_3^* (t) J_{3},\n\\end{aligned}  \\label{B}\n\\end{equation}%\nwhere $\\nu _{\\pm }(t),\\,\\,\\nu _{3}(t)$ may be complex functions of the time.\nHermitian invariants then can be easily built up as Hermitian combinations\nof $B$ and $B^{\\dagger }$. In particular if $B$ is a non-Hermitian invariant\nthe operator $I=B^{\\dagger }B- 1\/2$ \nis a Hermitian invariant, the fermion analog of the Lewis-Riesenfeld quadratic\ninvariant {\\normalsize \\cite{Lewis69}}.\n\nLet us note at this point that we look for FFO invariants as elements of the\nsame algebra $su(2)$ to which the Hamiltonian belongs. Similar is the approach\nused in \\cite{Chumakov86} in construction of invariants for\nthe nonstationary singular oscillator, where the related algebra is $su(1,1)$.\nThis is to be compared with the case of nonsingular oscillator, for which invariant\nladder operators have been built up as elements of the Heisenberg-Weyl algebra %\n$h_w$ (i.e. as linear combinations of coordinate and momentum operators $x$ and %\n$p$ \\cite{MMT70,Holz70}), while the related nonstationary Hamiltonian belongs to \n$su(1,1)$. %\nFor {\\it forced} boson oscillator the Hamiltonian belongs to the large algebra of \nsemi-direct sum $su(1,1)\\dot{+}h_w$ but the ladder operator invariants %\n are again elements of the invariant subalgebra $h_w$  %\n\\cite{Prants86,Dattoli86,Trif93}.\n\n\nTo proceed with construction of FFO invariants we substitute {\\normalsize (\\ref{B}) and \n(\\ref{Hf 2}) } into {\\normalsize (\\ref{19})}, and find the following system of\ndifferential equations for the parameter functions $\\nu_\\pm,\\,\\nu_3$: \\\n\\begin{eqnarray}\n\\dot{\\nu}_{3} &=&2\\ri(\\nu _{+}f^{\\ast }-\\nu _{-}f),  \\label{(a)} \\\\\n\\dot{\\nu}_{+} &=&\\ri(\\nu _{3}f-\\nu _{+}\\omega ),  \\label{(b)} \\\\\n\\dot{\\nu}_{-} &=&\\ri(\\nu _{-}\\omega -\\nu _{3}f^{\\ast }).  \\label{(c)}\n\\end{eqnarray}%\nSolutions to the above linear system of first order equations are\nuniquely determined by the initial conditions $\\nu _{\\pm }(0)=\\nu _{0,\\pm }$%\n, $\\nu _{3}(0)=\\nu _{0,3}$. If we want the invariants $B(t)$ and $B^{\\dagger\n}(t)$ be again fermion ladder operators, i.e. to obey the conditions\n\\begin{equation}\nB^{2}=0, \\quad \\{B,B^{\\dagger}=1,  \\label{constr}\n\\end{equation}%\nwe have to take $\\nu _{0,\\pm }$ and $\\nu _{0,3}$ satisfying\n\\begin{equation}\n \\nu _{0,3}^{2} = -4\\nu _{0,+}\\nu _{0,-},\\quad\n |\\nu _{0,-}|+|\\nu _{0,+}| = 1\\,. \\label{0}\n\\end{equation}%\nIndeed, for $B^{2}(t)$ and $\\{B(t),B^{\\dagger }(t)\\}$ we find\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle B^{2}=\\nu _{+}\\nu _{-}+\\tfrac{1}{4}\\nu _{3}^{2}\\equiv\n\\lambda _{1}$, \\\\[2mm]\n$\\displaystyle\\{B,B^{\\dagger}\\}=|\\nu _{-}|^{2}+|\\nu _{+}|^{2}+\\tfrac{1%\n}{2}|\\nu _{3}|^{2}\\equiv \\lambda _{2}$.%\n\\end{tabular}\n\\label{lam_i}\n\\end{equation}%\nThe quantities $\\lambda _{1}(\\nu _{\\pm },\\nu _{3})$, $\\lambda _{2}(\\nu _{\\pm\n},\\nu _{3})$ turned out to be two different '\\textit{constants of motion}'\nfor the system {\\normalsize (\\ref{(a)})}-{\\normalsize (\\ref{(c)})}, their\ntime derivatives being vanishing:\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle \n\\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\lambda _{1} = \\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\left( \\nu _{+}\\nu _{-}+\\tfrac{1%\n}{4}\\nu _{3}^{2}\\right) =0$, \\\\[3mm]\n$\\displaystyle\n\\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\lambda _{2} = \\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\left( |\\nu _{-}|^{2}+|\\nu\n_{+}|^{2}+\\tfrac{1}{2}|\\nu _{3}|^{2}\\right) =0$.%\n\\end{tabular}\n\\label{dotlam_i}\n\\end{equation}%\nTherefore we can fix the values of these constants as $\\lambda _{1}=0$, $%\n\\lambda _{2}=1$, i.e.\n\\begin{equation}\n\\begin{tabular}{l}\n$ \\displaystyle \\nu _{+}\\nu _{-}+\\tfrac{1}{4}\\nu _{3}^{2}=0,$\\\\[3mm]\n$\\displaystyle |\\nu _{-}|^{2}+|\\nu_{+}|^{2}+\\tfrac{1}{2}|\\nu _{3}|^{2}=1$, \n\\end{tabular}\n\\label{lam 0}\n\\end{equation}%\nand satisfy the conditions {\\normalsize (\\ref{constr})}. If furthermore the initial\nconditions are taken as\n\\begin{equation}\n\\nu _{-}(0)=1,\\,\\,\\,\\nu _{+}(0)=0=\\nu _{3}(0),  \\label{nu_i0}\n\\end{equation}%\nthen $B(0)=b$. Later on we work with these fermionic ladder operator invariants, %\ni.e. we consider conditions (\\ref{lam 0}) satisfied.\n\nLet us now recall that in the case boson nonstationary oscillator the ladder %\noperator invariants, constructed first in \\cite{MMT70, Holz70} (see also %\n\\cite{Chumakov86, Prants86}), are expressed in terms of one only parameter function %\n$\\epsilon(t)$, which obeys a simple second order equation, namely that of the %\nclassical oscillator with varying frequency. It turned out that this can be done %\nin the case of  fermionic oscillator as well.\nIn this aim we first  express all the three parameter functions $\\nu _{\\pm }(t)$,%\n $\\nu _{3}(t)$ in terms of one of them, which has to obey a second order %\ndifferential equation.\nLet for example, express $\\nu _{3}(t)$ and $\\nu _{-}(t)$ in terms of $\\nu _{+}(t)$%\n and its derivatives.  We have %\n\\begin{equation} \\label{nu3}\n\\nu _{3} =  - \\tfrac{\\ri}{f}(\\dot{\\nu}_{+} + \\ri \\nu _{+}\\omega ),\n\\end{equation}%\n\\begin{equation}\n\\nu _{-}=\\tfrac{1}{2f^{2}}\\left[ \\ddot{\\nu}_{+}+\\left( \\ri \\omega -\\tfrac{\\dot{f%\n}}{f}\\right) \\dot{\\nu}_{+}+\\left( 2ff^{\\ast }+\\ri\\dot{\\omega}-\\ri \\frac{\\omega }{f%\n}\\dot{f}\\right) \\nu _{+}\\right] .  \\label{nu-}\n\\end{equation}%\nSubstituting these expressions into the expression of $\\lambda_1$ in terms of $\\nu_\\pm,\n\\,\\nu_3$ and taking into account that $\\lambda_1$ is fixed to $0$ we find that $\\nu_+$ %\nshould satisfy the following second order equation,\n\n\\begin{equation}\n2\\nu _{+}\\ddot{\\nu}_{+}-\\dot{\\nu}_{+}^{2} - 2\\nu_{+}\\dot{\\nu}_{+}\\tfrac{\\dot{f}}{f} +\n4\\nu _{+}^{2}\\left(\n|f|^{2}+\\tfrac{\\omega ^{2}}{4}+ \\ri\\tfrac{\\dot{\\omega}}{2}-\\ri \\tfrac{\\omega %\n\\dot{f }}{2f}\\right)  =  0 .  \\label{lam}\n\\end{equation}%\n Using this, and supposing that $\\nu_{+}\\neq 0$, we obtain for $\\nu _{-}$ a more %\ncompact expression in terms of $\\nu _{+}$ and $\\dot{\\nu}_{+}$, %\n\\begin{equation}\n\\nu _{-}= -\\nu _{3}^{2}\/4\\nu _{+},  \\label{nu- 2}\n\\end{equation}%\nwhere $\\nu_3$ is given again by eq. (\\ref{nu3}).\n\nThus the operators $B(t),\\,B^{\\dagger}(t)$, eq. (\\ref{B}), are fermionic %\nladder operator invariants for the forced oscillator (\\ref{Hf}), (\\ref{Hf 2})%\nif  $\\nu _{3}$ and $\\nu _{-}$ are given by eqs. (\\ref{nu3}) and (\\ref{nu- 2}),%\nand $\\nu _{+}(t)$ is a nonvanishing solution of the second order equation (\\ref{lam}).\n\nNext we try to linearize the auxiliary eq. (\\ref{lam}). In this purpose we put\n\\begin{equation}\n\\nu _{+}(t) = \\tfrac{1}{2}{\\epsilon'} ^{2}(t)  \\label{nu+ eps'}\n\\end{equation}%\nand obtain that  $\\epsilon' (t)$ satisfies the linear equation\n\\begin{equation} \\label{eps' eq}\n\\ddot{\\epsilon}' - \\tfrac{\\dot{f}}{f}\\dot{\\epsilon}'+\\Omega' (t)\\epsilon' =0,\n\\end{equation}%\nwhere\n\\begin{equation}\n\\Omega' (t)=|f(t)|^{2}+\\tfrac{1}{4}\\omega ^{2}(t)+\\tfrac{\\ri}{2}\\dot{\\omega}-%\n\\tfrac{\\ri}{2}\\omega \\tfrac{\\dot{f}}{f}.\n\\end{equation}\n\nIn terms of $\\epsilon'$ the formulas (\\ref{nu-}) and (\\ref{nu3} for $\\nu_-,\\, \\nu_3$ read\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle\\nu _{-}=-\\frac{1}{2\\epsilon'}\\nu_3^{2}$, \\\\[2mm]\n$\\displaystyle\\nu _{3}=\\frac{1}{f}\\left( \\tfrac{\\omega }{2}{\\epsilon'}\n^{2}-\\ri\\epsilon' \\dot{\\epsilon}'\\right) $.%\n\\end{tabular}%\n\\end{equation}%\nThe term in (\\ref{eps' eq}) proportional to the first derivative can be eliminated %\nby the substitution %\n\\begin{equation}\\label{eps}\n\\epsilon'  = \\epsilon \\exp \\left( \\tfrac{1}{2}\\tint_{0}^{t}{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}\\tau \\,\\dot{f}%\n(\\tau )\/f(\\tau )\\right) .\n\\end{equation}%\nThis leads to the desired simple equation for $\\epsilon$,\n\\begin{equation}\\label{eps eq}\n\\ddot{\\epsilon}+\\Omega (t)\\epsilon = 0,\n\\end{equation}%\nwhere $ \\Omega(t)= \\Omega^{\\prime }(t)  + \\ddot{f}\/2f - 3\\dot{f}\\,^{2}\/4f^{2}$. %\nEquation (\\ref{eps eq}) is of the same type, as the auxiliary equation used in the %\ncase of nonstationary boson oscillator \\cite{MMT70, Holz70}. Here the %\n'squared frequency' $\\Omega$ however is complex and depends in a different manner %\non the corresponding Hamiltonian parameters. And the solutions are subject to %\ndifferent constraints, stemming from the different commutation relations: \nin terms of our $\\epsilon$, eq. (\\ref{eps eq}), the constraint $\\lambda_2=1$ reads %\n( $\\epsilon'$ is related to $\\epsilon$ according to (\\ref{eps})),\n\\begin{equation}\n\\tfrac{|\\epsilon' |^{4}}{4}\\left( 1+\\tfrac{2}{|f|^{2}}\\left\\vert\n\\tfrac{\\omega }{2}\\epsilon' -\\ri\\dot{\\epsilon}'\\right\\vert ^{2}+\\tfrac{1}{|f|^{4}%\n}\\left\\vert \\tfrac{\\omega }{2}\\epsilon' -\\ri\\dot{\\epsilon}'\\right\\vert^{4}\\right) =1 ,\n\\end{equation}%\n while in  the boson case the constraint is  ${\\rm Im}\\left({\\epsilon}^*\\dot{\\epsilon}\\right) = 1$ \n\\cite{MMT70, Holz70}.\n\nTo finalize this section let note that in the particular case of the %\n\\textit{free} fermion oscillator, $f(t)\\equiv 0$, the explicit solutions of the problem can %\n be easily found in the form %\n\\begin{equation}\n\\begin{tabular}{l}\n$\\displaystyle \n\\nu _{\\pm }(t)=\\nu _{0,\\pm }\\re^{\\pm \\ri\\tint^{t}\\omega (\\tau ){\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}\\tau },$\\\\\n$ \\displaystyle \n\\nu_{3}=\\nu _{0,3},$\n\\end{tabular}\n\\label{solutions1}\n\\end{equation}%\nwhere $\\nu _{0,\\pm }$, $\\nu _{0,3}$ are constants. To ensure the fermionic %\ncommutation relations of $B(t),\\,B^\\dagger(t)$ they have to obey the relations\n$\\nu _{0,-}\\,\\nu _{0,+} + \\nu _{0,3}^{2}\/4 =0$ \\,\\, and \\,\\, $ |\\nu\n_{0,-}|^{2} + |\\nu _{0,+}|^{2} + |\\nu _{0,3}|^{2}\/2  = 1.$\n\\medskip\n\n\\section{CS for the fermion forced oscillator}\n\nWe define coherent states (CS) for a given fermion system as eigenstates of\nthe corresponding invariant fermion annihilation (or creation) operator $B(t)\n$. Since the most general fermion one mode Hamiltonian operator is of the\nform of (nonstationary) forced oscillator (\\ref{Hf 2}), the one-mode fermion\nCS are defined as eigenstates of the invariant ladder operator $B(t)$\n(eqs. (\\ref{B}), (\\ref{constr})):\n\\begin{equation}\nB(t)|\\zeta ;t\\rangle =\\zeta |\\zeta ;t\\rangle .  \\label{|z;t> 1}\n\\end{equation}%\nSince $B(t)$ is invariant operator,  the eigenvalue $\\zeta $ does\nnot depend on time $t$. In terms of the $\\zeta $, $B(t)$, $B^{\\dagger }(t)$\nand the $B(t)$-vacuum $|0;t\\rangle $ we have for $|\\zeta ;t\\rangle $ the\nsame formulas as for the canonical fermion CS $|\\zeta \\rangle $ which are\ndefined {\\normalsize \\cite{Klauder, Abe89, Maam92, Junker98, Cahill99}} as\n\\begin{equation}\n\\left\\vert \\zeta \\right\\rangle =e^{-\\frac{1}{2}\\zeta ^{\\ast }\\zeta }\\left(\n\\left\\vert 0\\right\\rangle -\\text{ }\\zeta \\left\\vert 1\\right\\rangle \\right)\n\\,.  \\label{|z>}\n\\end{equation}%\nwhere the eigenvalue $\\zeta $ is a Grassmannian variable: $\\zeta ^{2}=0,\\\n\\zeta \\zeta ^{\\ast }+\\zeta ^{\\ast }\\zeta =0$, $\\left\\vert 0\\right\\rangle $\nis the fermionic vacuum, $b\\left\\vert 0\\right\\rangle =0$, and $\\left\\vert\n1\\right\\rangle $ is the one-fermion state, $\\left\\vert 1\\right\\rangle\n=b^{\\dagger }\\left\\vert 0\\right\\rangle $. In particular\n\\begin{equation}\n|\\zeta ;t\\rangle = \\re^{-\\tfrac{1}{2}\\zeta ^{\\ast }\\zeta }\\left( |0;t\\rangle\n-\\zeta B^{\\dagger }(t)|0;t\\rangle \\right) .  \\label{|z;t> 2}\n\\end{equation}%\nIt remains therefore to construct the (normalized) new ground state $%\n|0;t\\rangle $ according to its defining equations\n\\begin{equation}\n\\begin{aligned} B(t) |0;t\\rangle = 0 ,\\quad \n\\ri\\frac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{dt} |0;t\\rangle = H_{f}|0;t\\rangle. \\end{aligned}  \\label{|0;t> 1}\n\\end{equation}%\nWe put\n\\begin{equation}\n|0;t\\rangle =\\alpha _{0}(t)|0\\rangle +\\alpha _{1}(t)|1\\rangle ,\n\\label{|0;t> 2}\n\\end{equation}%\nsubstitute this into (\\ref{|0;t> 1}) and after some tedious calculations find%\n\\begin{eqnarray}\n\\alpha _{1}(t) &=&\\alpha _{0}(t)\\frac{\\nu _{3}^{\\ast }(t)}{2\\nu _{+}^{\\ast\n}(t)}, \\\\\n\\alpha _{0}(t) &=&\\sqrt{|\\nu _{+}(t)|}\\exp \\left[ -\\tfrac{\\ri}{2}\\left(\n\\varphi _{\\nu _{+}}(t)+\\tint_{0}^{t}(2g(\\tau )+\\omega (\\tau )){\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}\\tau \\right) %\n\\right] ,\n\\end{eqnarray}%\nwhere $\\varphi _{\\nu _{+}}$ is the phase of $\\nu _{+}(t)$. The state $|\\zeta\n;t\\rangle $ will represent the exact time evolution of an initial canonical\nCS $|\\zeta \\rangle $ if the initial conditions (\\ref{nu_i0}) are imposed: %\n$|\\zeta ;0\\rangle = |\\zeta \\rangle $. In this case, the time evolved state %\n$ |\\zeta ;t\\rangle $ could be again an eigenstate of $b$ if the oscillator %\nis not 'forced', i.e. if $f(t)=0$.\nLet us note that the time-dependence of the constructed states is obtained %\nin terms of solutions to the system of auxiliary equations (\\ref{(a)})-(\\ref{(c)}),%\nor equivalently to the 'classical oscillator' equation (\\ref{eps eq}).\n\nOur method of construction of dynamical invariants differs slightly from the\nLewis-Riesenfeld method \\cite{Lewis69} (developed for bosonic oscillators).\nLewis and Riesenfeld used to first construct Hermitian invariant, which then\nis represented as a product of normally ordered ladder operators. To make\nconnection to their approach let us suppose that we first succeeded to\nconstruct the Hermitian invariant $N(t)$ and to find some ladder operators $%\n\\tilde{B}(t)$, $\\tilde{B}^{\\dagger }(t)$ that factorize it: $N(t)=\\tilde{B}%\n^{\\dagger }(t)\\tilde{B}(t)$. It is clear that $\\tilde{B}(t)$ may differ from\nour non-Hermitian invariant $B(t)$ in a phase factor:\\thinspace\\ $\\tilde{B}%\n(t) = {\\rm e}^{\\ri\\varphi (t)}B(t)$.\\newline\nWe can then in a standard way construct normalized eigenstates of $N(t)$,\n\\begin{equation}\nN(t)\\widetilde{|0;t\\rangle }=0,\\quad N(t)\\widetilde{%\n|1;t\\rangle }=\\widetilde{|1;t\\rangle },  \\label{tld 2}\n\\end{equation}%\nand of $\\tilde{B}(t)$, \\\n\\begin{equation}\n\\tilde{B}(t)\\widetilde{|\\zeta ;t\\rangle }=\\zeta \\widetilde{|\\zeta ;t\\rangle },\n\\end{equation}%\n\\begin{equation}\n\\widetilde{|\\zeta ;t\\rangle }=\\left( 1-\\tfrac{1}{2}\\zeta ^{\\ast }\\zeta\n\\right) \\left[ \\widetilde{|0;t\\rangle }-\\zeta \\widetilde{|1;t\\rangle }\\right]\n\\label{tld 3}\n\\end{equation}%\nwhich however do not obey the Schr\\\"odinger equation since, in general $ \\tilde{B}(t)$\nmay not be invariant. To obtain solutions $|n;t\\rangle $ and $ |\\zeta ;t\\rangle $ the\nabove eigenstates $\\widetilde{|n;t\\rangle }$, $n=0,1$, should also be multiplied by\nphase factors,\n\\begin{equation}\n|n;t\\rangle = {\\rm e}^{\\ri\\phi _{n}(t)}\\widetilde{|n;t\\rangle },\\quad n=0,1,\n\\label{tld 4}\n\\end{equation}%\n\\begin{equation}\n|\\zeta ;t\\rangle =\\left( 1-\\tfrac{1}{2}\\zeta ^{\\ast }\\zeta \\right) \\left[\n{\\rm e}^{\\ri\\phi _{0}(t)}\\widetilde{|0;t\\rangle }-\\zeta {\\rm e}^{\\ri\\phi _{1}(t)}%\n\\widetilde{|1;t\\rangle }\\right]  \\label{tld 5}\n\\end{equation}%\nwhich should obey the equations \\\n\\begin{equation}\n\\tfrac{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e}}{{\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t}\\phi _{n}=\\widetilde{\\langle n;t|}\\ri\\tfrac{\\partial }{\\partial t}%\n-H\\widetilde{|n;t\\rangle }.  \\label{tld 6}\n\\end{equation}%\nEvidently the state (\\ref{tld 5}) is an eigenstate of $\\tilde{B}(t)$ with time\ndependent eigenvalue $\\zeta (t)=\\zeta \\exp (\\ri\\varphi (t))$, $\\varphi\n(t)=\\phi _{1}(t)-\\phi _{0}(t)$.\\newline\nThe phase $\\varphi (t)=\\phi _{1}(t)-\\phi _{0}(t)$ consists of two parts -\ngeometrical one $\\varphi ^{G}$, and dynamical one $\\varphi ^{D}=\\varphi\n-\\varphi ^{G}$ \\cite{Maam99},\n\\begin{eqnarray}\n\\varphi ^{G}(t)  &=&\\varphi (t)+\\int_{0}^{t}\\left( \\widetilde{\\langle 1;t^{\\prime }|}H%\n\\widetilde{|1;t^{\\prime }\\rangle }-\\widetilde{\\langle 0;t^{\\prime }|}H%\n\\widetilde{|0;t^{\\prime }\\rangle }\\right) {\\rm d}}   \\def\\ri{{\\rm i}} \\def\\re{{\\rm e} t^{\\prime }.\n\\end{eqnarray}\n\n\\subsection*{Concluding Remarks}\n\n\\ \\ \\ In this article, we have studied fermionic system of nonstationary\nforced oscillator and we have constructed invariant ladder operators and the\nrelated Fock and coherent states. We succeeded to express these invariants\nand the time evolution of the corresponding states in terms of the same\nclassical equation, that describes the evolution of coherent states of the\nboson nonstationary (forced) oscillator \\cite{MMT70, Holz70}. The relation of\nthe invariant ladder operators method to the Lewis-Riesenfeld method \\cite{Lewis69}\nwas briefly described on the example of nonstationary fermion systems. \\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}INTRODUCTION}\n\nIt is well recognized that the $\\alpha$ particle interaction with atomic nuclei is important in astrophysics \\cite{1}. Even if astrophysical reactions involving helium do not proceed through the strong $\\alpha$-cluster states (because of their high excitation energy), these states can provide $\\alpha$ width to the states that are closer to the region of astrophysical interest through configuration mixing.\n\nFor a long time, the surprising alpha cluster structure has been a stimulus for the development of classical shell model approaches (see \\cite{2} for new results). Additionally, work by the authors of Ref. \\cite{3} recently ``strengthened the theoretical motivation for experimental searches of alpha cluster states in alpha-like nuclei\" \\cite{3}. The authors of Ref. \\cite{3} related the nuclear structure in light systems of even and equal numbers of protons and neutrons with the first-order transition at zero temperature from a Bose-condensed gas of alpha particles (\\textsuperscript{4}He nuclei) to a nuclear liquid.\n\n\\textsuperscript{20}Ne nucleus presents a famous example of the manifestation of the alpha-cluster structure, and therefore it makes this nucleus a touchstone for \\textit{ab initio} approaches \\cite{4}. The nucleus \\textsuperscript{20}Ne is a benchmark case for the traditional shell model and its extension into algebraic and clustering domains. The well-established effective interaction Hamiltonians such as \\cite{5} not only shows an outstanding agreement with experimental data for \\textit{sd}-shell nuclei but also generates configuration mixing that shows transition to deformation and clustering.\n\nThe remarkable feature of \\textsuperscript{20}Ne nucleus shows up in the fact that almost all the observed states below 10 MeV can be classified into several overlapping rotational-like bands with the first one based on the ground  0$^+_1$ state. There are three other bands based on 0$^+$ levels: on 0$^+_2$ at 6.725 MeV, on a very narrow 0$^+_3$ at 7.191 MeV, on a very broad 0$^+_4$ at $\\sim$ 8.7 MeV \\cite{6} which are of evident cluster structure. The 0$^+_2$ and 0$^+_4$ bands have $\\alpha$+\\textsuperscript{16}O core structure as can be seen from their reduced $\\alpha$ particles widths, and probably the 0$^+_3$ band has predominant \\textsuperscript{12}C+\\textsuperscript{8}Be structure which manifests used itself in the selectivity of the \\textsuperscript{8}Be transfer reactions \\cite{7}. As the ground state band and the 0$^+_2$ band in \\textsuperscript{20}Ne can be related with similar structures in \\textsuperscript{16}O and \\textsuperscript{12}C, the ``additional\" structure of 0$^+_4$ states is not understood \\cite{4}. The cluster approaches \\cite{8} related the 0$^+_4$ band with large $\\alpha$-widths, starting with the so-called ``\\textsuperscript{16}O+$\\alpha$\" higher nodal band, which has one more nodal point in ``\\textsuperscript{16}O+$\\alpha$\" relative wave function than the lower bands have. However it appeared that there are too many bands with a similar structure.\n\nThe $\\alpha$ particle decay threshold in \\textsuperscript{20}Ne is 4.73 MeV, while the threshold for proton decay is at 12.8 MeV (neutron decay threshold is even higher). Therefore resonance $\\alpha$ particle scattering should be considered as an evident way to obtain data on the natural parity levels in \\textsuperscript{20}Ne up to 13 MeV excitation energy. Indeed the majority of adopted data \\cite{6} on level properties in the region in question based on a resonance work and an analysis made in 1960 year. More recently the $\\alpha$+\\textsuperscript{16}O resonance scattering experiment were developed further to backward angles and the data were reanalyzed using \\textit{R} matrix code Multi 6 \\cite{9}. The authors \\cite{9} obtained quite different results from those used in for many levels (see Table 1); in particular the broad 0$^+_4$ and 2$^+_4$ levels appeared even much more broader. The authors \\cite{9} noted difficulties of the fit in the region of 6-8 MeV excitation energies, in a region mainly free from narrow resonances. Evidently strong states of over 1 MeV width should influence a very broad excitation region (see for instance \\cite{10}).\n\nT. Fortune et al. \\cite{11,12} were the first who recognized the importance of the fact that single particle structure of the broad states is in drastic contradiction with the shell model predictions. They \\cite{11,12} also proposed the idea of mixing different configurations to explain the effect. The same idea was used in \\cite{13}. However, the authors of Refs. \\cite{11,12} used old data for the 0$^+_2$ state and used some estimates for the properties of the broad states proposed in Ref. \\cite{14} to support the idea. Later measurements \\cite{6} gave the width of 19 keV for the 6.72 MeV state (which is about 25\\% larger than that used in work \\cite{11}).\n\n\\begin{figure}[!t]\n    \\begin{center}\n    \\include{fig1}\n    \\end{center}\n    \\caption{E-T spectrum for the zero degrees detector. Alpha particles dominate; one can see also a weaker proton locus below the alpha particles.}\n\\end{figure}\n\nThe experimental aim of this work is to obtain new information on the structure of \\textsuperscript{20}Ne states, especially the broad 0$^+$, 2$^+$ states. Unlike other experimentalists, we used the Thick Target Inverse Kinematics method (TTIK) (see \\cite{15,16,17,18,19} and references therein) to study the excitation functions for the \\textsuperscript{16}O($\\alpha$, $\\alpha$)\\textsuperscript{16}O elastic scattering in the \\textsuperscript{20}Ne excitation region of 5.5-9.6 MeV and in a broad angular interval. On the theoretical side, we also used multi configuration shell model calculation to understand the limits of this approach in a description of the cluster states.\n\n\\section{\\label{sec:level2}EXPERIMENT}\n\n\\begin{figure}[!t]\n    \\begin{center}\n    \\include{fig2}\n    \\end{center}\n    \\caption{The \\textsuperscript{16}O($\\alpha$, $\\alpha$)\\textsuperscript{16}O elastic scattering excitation function  at 180$^\\circ$ cm. The excitation energies in \\textsuperscript{20}Ne, E$_x$ in Table 1,  are related with cm energy, E$_{cm}$, by expression,  E$_x$=E$_{cm}$+4.73 MeV.  The bold (red) line is the \\textit{R} matrix fit with the parameters of the present work. The dot (cyan) line is a fit with the 0$^+_4$ excitation energy of 8.3 MeV \\cite{11}, and dot (black) line is a fit with the 0$^+_4$ energy excitation energy of 8.62 MeV and the width of 1.472MeV \\cite{9}.}\n\\end{figure}\n\nThe experiment was performed at the DC-60 cyclotron (Astana) \\cite{17} which can accelerate heavy ions up to the 1.9 MeV\/A energy. While the TTIK method can't compete with a classical approach in terms of energy resolution, the possibility of observing excitation functions at and close to 180 degrees, where the resonance scattering dominates over the potential scattering, enables one to obtain more reliable information on the broad states.  In the TTIK technique the inverse kinematics is used; and the incoming ions are slowed in a helium target gas. The light recoils, $\\alpha$ particles, are detected from a scattering event. These recoils emerge from the interaction with the beam ions and hit a Si detector array located at forward angles while the beam ions are stopped in the gas, as $\\alpha$ particles have smaller energy losses than the scattered ions. The TTIK approach provides a continuous excitation function as a result of the slowing down of the beam.\n\nFor the present experiment, the scattering chamber was filled with helium of 99.99\\% purity. The 30 MeV \\textsuperscript{16}O beam entered the scattering chamber through a thin entrance window made of 2.0 $\\mu$m Ti foil. Eight monitor Si detectors were placed in the chamber to detect \\textsuperscript{16}O ions elastically scattered from the Ti foil at 21$^\\circ$ angle. This array monitors the intensity of the beam with precision better than 4\\%. Fifteen 10x10 mm$^2$ Si detectors were placed at a distance of $\\sim$ 500 mm from the entrance window in the forward hemisphere at different laboratory angles starting from zero degrees. The gas pressure was chosen to stop the beam at distance of 40 mm from the zero degrees detector. The detector energy calibration and resolution ($\\sim$ 30 keV) were tested with a \\textsuperscript{226}Ra, \\textsuperscript{222}Rn, \\textsuperscript{218}Po and \\textsuperscript{214}Po $\\alpha$-source. The experimental set up was similar to that used before \\cite{19}, and more details can be found in Ref. \\cite{17,19}. The main errors in the present experimental approach are related to the uncertainties of the beam energy loss in the gas. To test the energy loss, we placed a thin Ti foil (2.0 $\\mu$m) at different distances from the entrance window. This can be used during the experiment without cycling vacuum. We found that the data \\cite{20} for energy loss of \\textsuperscript{16}O in helium are correct. The details of these tests will be published elsewhere. As a result, we estimated that the uncertainties in the absolute cross section are less than 6\\%. This conclusion was tested by comparison with the Rutherford cross sections at low energies. The agreement with the Rutherford scattering is within 5\\% error bars at all angles (see Fig.3).\n\nTogether with the amplitude signal, the Si detectors provided for a fast signal. This signal together with a ``start\" signal from RF of the cyclotron was used for the Time-of-Flight measurements. This E-T combination is used for particle identification in the TTIK approach \\cite{16,17,18}. Of course, only $\\alpha$ particles should be detected as a result of the interaction of \\textsuperscript{16}O with helium at the chosen conditions. However, protons can be created in the Ti window, and protons can appear due to hydrogen admixtures in the gas. Indeed, we have observed a weak proton banana, likely as a result of  reactions in the window. These protons were easily identified by TF and separated from the $\\alpha$ particles, as  seen in Fig.1.\n\n\\section{\\label{sec:level3}Experimental Results and Discussion}\n\n\\begin{figure}[!t]\n    \\begin{center}\n    \\include{fig3}\n    \\caption{\\textit{R} matrix fit (bold red curve) of the excitation functions for the $\\alpha$+\\textsuperscript{16}O elastic scattering. (a) The dashed (cyan) curve presents the data of  the 0$^+$ level at the 8.3 MeV excitation energy \\cite{11} and (e) dot (black) line is a fit with 2$^+$ level at the excitation energy of 9.0 MeV \\cite{6}.}\n    \\end{center}\n\\end{figure}   \n\n\\begin{table*}[!t]\n\\label{tab:1}\n\\caption{\\textsuperscript{20}Ne levels}\n\\begin{center}\n\\begin{tabular}{lccccccccccclccl}\n\\hline\n\\hline\n\\multirow{1}{*}{N} & \\multicolumn{3}{c}{TUNL data \\cite{6}} & \\multirow{1}{*}{} & %\n    \\multicolumn{2}{c}{H. Shen et al. \\cite{9}} & \\multirow{1}{*}{} & %\n    \\multicolumn{3}{c}{This work} & \\multirow{1}{*}{} & \\multicolumn{4}{c}{CNCIM} \\\\\n\\cline{2-4}\\cline{6-7}\\cline{9-11}\\cline{13-16}\n & E$_x$ & J$^\\pi$ & $\\Gamma_\\alpha$ & & E$_x$ & $\\Gamma_\\alpha$ & & E$_x$ &\n $\\Gamma_\\alpha$ & $\\gamma_\\alpha$ & & E$_x$ & J$^\\pi$ & SF$_p$ & SF$_\\alpha$\\\\\n & (MeV) & & (keV) & & (MeV) & (keV) & & (MeV) & (keV) & & & (MeV) &  &  & \\\\\n\\hline\n\n1 &\t0 & 0$^+_1$ &  & & - & - & & 0 & & Large & & 0 & 0$^+$ & 0.36 & 0.73\\\\\n2 &\t1.63 & 2$^+_1$ & &  & - & - & & 1.63 & & Large & & 2.242 & 2$^+$ & 0.41 & 0.67\\\\\n3 &\t4.25 & 4$^+_1$ & &  & - & - & & 4.25 & & Large & & 4.58 & 4$^+$ & & 0.62\\\\\n\n4 &\t5.78 & 1$^-$ & (28$\\pm$3)x10$^{-3}$ &  & - & - &  & 4.45 & 0.03 & 1.4\\\\\t\t\t\t\n5 &\t6.73 & 0$^+_2$ & 19$\\pm$0.9 & & 6.72 & 11 & & 6.78 & 20.6 & 0.47 & & 6.94 & 0$^+_3$ & 0.55 & 0.46\\\\\n6 &\t7.16 & 3$^-$ & 8.2$\\pm$0.3 & & 7.16 & 10 & & 7.18 & 8.3 & 1.37\\\\\t\t\t\t\n7 &\t7.19 & 0$^+_3$ & 3.4$\\pm$0.2 & & 7.19 & 5 & & 7.20 & 3\t& 0.019 & & 6.27** & 0$^+_2$ & 0.055 & 0.44**\\\\\n8 &\t7.42 & 2$^+_2$ & 15.1$\\pm$0.7 & & 7.43 & 7 & & 7.44 & 14.3 & 0.19 & & 7.39 & 2$^+_3$ & 0.01 & 0.12\\\\\n9 &\t7.83 & 2$^+_3$ & 2 & & 7.83 & 1 & & 7.85 & 3.68 & 0.01 & & 7.15** & 2$^+_2$ & 0.12 & 0.18**\\\\\n\n10 & 8.45 & 5$^-$ & 0.013$\\pm$0.004 & & 8.45 & 0.02 & & 8.45 & 0.013\\\\\t\t\t\t\t\n11 & 8.71 & 1$^-$ & 2.1$\\pm$0.8 & & & & & 8.71 & 3.5\\\\\t\t\t\t\t\n12 & $\\approx$8.7 & 0$^+_4$ & $>$800 & & 8.62 & 1470 & & 8.77$\\pm$0.15 & 750$\\pm$220 & $\\sim$0.25 & & 9.66** & 0$^+_4$ & 0.002 & 0.18**\\\\\n13 & 8.78 & 6$^+_1$ & 0.11$\\pm$0.02 & & - & - & & 8.78 & 0.14 & 0.5 & & 9.49 & 6$^+$ & & 0.51\\\\\n14 & 8.85 & 1$^-$ & 19 & & 8.84 & 27 & & 8.85 & 18.0\\\\\t\t\t\t\t\n15 & 9.00 & 2$^+_4$ & $\\approx$800 & & 8.87 & 1250 & & 8.79$\\pm$0.10 & 695$\\pm$120 & 0.86 & & 8.36** & 2$^+_4$ & 0.02 & 0.02**\\\\\n16 & 9.03 & 4$^+_3$ & 3 & & 9.02 & & & 9.03 & 1.9 & 0.03 & & 9.0 & 4$^+$ & & 0.09\\\\\n\n17 & 9.12 & 3$^-$ & 3.2 & & 9.09 & 4 & & 9.13 & 4.1\\\\\n18 & 9.19 & 2$^+$ &  &  & - & - & & (9.29) & $\\leq$10\t\\\\\t\t\t\t\n19 & 9.48 & 2$^+$ & 29$\\pm$15 & & 9.48 & 46 & & 9.48 & 65$\\pm$20 & 0.02?\\\\\t\t\t\t\n20 & 9.99 & 4$^+_4$ & 155$\\pm$30 & & 10.02 & 150 & & 9.97 & 157 & 0.38 & & 9.5 & 4$^+$ & & 0.009\\\\\n21* & 10.26 & 5$^-$ & 145$\\pm$40 & & 10.26 & 190 & & 10.26 &\t& 1.9\t\\\\\t\t\t\n22 & 10.41 & 3$^-$ & 80 & & 10.40 & 101 & & 10.41 & &\t\\\\\n\n23 & 10.58 & 2$^+$ & 24 & & 10.56 & 15 & & 10.58 & & & & 10.2 & 2+ & 0.005 & 0.04\\\\\n24 & 10.80 & 4$^+_4$ & 350 & & 10.75 & 400 & & 10.80 & & & & 10.7 & 4$^+$ & & 0.04\\\\\n25 & 10.97 & 0$^+_5$ & 580 & & 10.99 & 700 & & 10.97 & & & & 11.9 & 0$^+$\t\\\\\t\n26 & 11.24 & 1$^-$ & 175 & & 11.19 & 85 & & 11.24 & &\\\\\t\t\t\t\t\n27 & 11.95 & 8$^+$ &  (3.5$\\pm$1.0)x10$^{-2}$  & & & & & 11.95 & & 0.35 & & 11.50 & 8$^+$ & & 0.40\\\\\n\\hline\n\\hline\n\\multicolumn{16}{l}{ * For the levels with numbers 21-27 the parameters of the present fit were fixed as in \\cite{6} } \\\\\n\\multicolumn{16}{l}{ ** Calculated in \\textit{psd} space. SF is to the first excited state in \\textsuperscript{16}O; SFs for the ground state in \\textsuperscript{16}O are $\\leq$ 0.1 } \\\\\n\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe experimental excitation functions were analyzed using multilevel multichannel \\textit{R} matrix code \\cite{21}. The calculated curves were convoluted with the experimental energy resolution. The experimental energy resolution was $\\sim$ 30 keV at zero degrees and deteriorated up to $\\sim$100 keV with angles estranging from zero degrees. We did not notice a deterioration of the energy resolution with the energy loss of the beam in the chamber. As it seen in Table 1 the excitation energies of the resonances of the present work agree with the adopted ones \\cite{6} within 10-15 keV. This agreement is an evidence of our overall good energy calibrations and the correct account of the ion energy loss in helium in the present work. Fig.2 and Fig.3 give the experimental excitation functions together with the present \\textit{R} matrix fit. Fig.2 demonstrates the data at 180$^\\circ$ and illustrates the differences in the fits due to different parameters of the broad 0$^+_4$ resonance. The data on the resonances used in the present \\textit{R}-matrix fit are summarized in the Table 1 together with the adopted data \\cite{6}. Data of the last \\textit{R} matrix analysis \\cite{9} are also given in Table 1. The analysis \\cite{9} resulted in level parameters which are often different from the adopted.\n\nOur analysis (Table 1) resulted in  small discrepancies with the data \\cite{6} in the detail description of the narrow states (widths less than 10 keV). The \\textit{R}-matrix code \\cite{21} is tuned for the TTIK measurements and for the analyses of states with a width of over 10 keV to accelerate automatic fit calculations. Therefore the small disagreements for the narrow states are not significant. We focused on the broader states and on the part of the excitation function changing slowly with energy and angle.\n\nOur analysis indicates that all strong alpha cluster states at 5$\\sim$6 MeV below or up the investigated excitation energy region can influence the \\textit{R}-matrix fit. Therefore we included in the fit the \\textsuperscript{20}Ne ground state, the first 2$^+$ and 4$^+$ states and 1$^-$ (5.67 MeV) state in the fit, even though  below the investigated region (see Table 1). Among these states, only the 1$^-$ state is above the $\\alpha$ particle decay threshold in \\textsuperscript{20}Ne; the reduced width of this state is known, and it is large. Shell model calculations also give very large spectroscopic factors for all members of the ground state band. Indeed, a good agreement needs large values of the corresponding amplitudes (over 0.7). Above the investigated excitation energy region, high spin $\\alpha$-cluster resonances are mainly known. Each of these resonances (see Table 1) considered separately influences the fit, especially at 180$^\\circ$. However, their joint influence is much weaker. This cancellation is due to different parities of the spins. Only the influence of the closest to the investigated region, the 4$^+$ (9.99 MeV) resonance, can be noticed. A somewhat better fit needs the width of this resonance to be slightly larger $\\sim$160 keV (well within the quoted uncertainties, see Table 1). Parameters of all other resonances above the investigated region were fixed according to the data of Ref. \\cite{6}. A good general fit ($\\chi^2$=1.1) was reached in this way without any backward resonance inclusion.\n\n\\begin{table*}[!t]\n    \\caption{$\\alpha$+\\textsuperscript{12}C levels in \\textsuperscript{16}O}\n    \\centering\n    \\begin{tabular}{ccccccccc}\n    \\hline\n    \\hline    \n    \\multicolumn{1}{c}{\\textsuperscript{16}O level} & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\Gamma_\\alpha$ $_{exp}$ keV \\cite{6} } & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\gamma_\\alpha$(1); -V$_0$ MeV} & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\gamma_\\alpha$(2); -V$_0$ MeV} & \\multirow{1}{*}{} & \\multicolumn{1}{c}{$\\gamma_\\alpha$(3); -V$_0$ MeV}\\\\ \n    \\cline{1-1}\\cline{3-3}\\cline{5-5}\\cline{7-7}\\cline{9-9}\n    1$^-$; 9.58 MeV & & 420$\\pm$20 & & 0.70; 138.2 & & 0.72; 150.0 & & 0.84; 158.5 \\\\\n    4$^+$; 10.36 MeV & & 26$\\pm$3 & & 0.68; 125.3 & & 0.88; 139,6 & & 1.21; 143.2 \\\\\n    \\hline\n    \\hline    \n    \\end{tabular}\n    \\label{tab:2}\n\\end{table*}\n\nAll resonances are at the maximum at 180$^{\\circ}$. The broad hump at this angle at cm energy of 4 MeV (Fig. 2) is a clear indication for the presence of low spin states, 0$^+$ and 2$^+$. Higher spin states are narrower. A single level (2$^+$) cannot produce the strong peak at different angles, and levels of different parity, such as 2$^+$ and 1$^-$, interfere destructively at 180$^{\\circ}$. The present analysis resulted in two practically degenerate states at $\\sim$ 8.8 MeV with the same width (see Table 1). Our results for the broad 2$^+$ are rather close to the adopted values. However, if this level is moved to 9.0 MeV excitation energy (as in \\cite{6}) then the fit becomes worse, especially in the vicinity of the dip die to the presence of another 2$^+$ level at 9.48 MeV (Fig. 3(e)).  The fit becomes even worse with a very broad 2$^+$ level resulting from the parameters of Ref. \\cite{9}.\n\nT. Fortune at al., \\cite{11} observed a broad distribution with a center at 8.3 MeV excitation energy in \\textsuperscript{20}Ne and related it with the 0$^+$ level. Fig. 2 presents \\textit{R} matrix calculations with the 0$^+$ level to be moved to 8.3 MeV excitation energy. This move destroyed the good fit. Fig.2 shows also that a very broad 0$^+$ level of the fit \\cite{9} destroys the agreement.\n\nThe 18$^{th}$ (2$^+$) level of \\cite{6} is in the energy region of the present investigation. It was observed in a single work in a study of \\textsuperscript{20}Na $\\beta^+$ decay. We have not found any reliable evidence for the presence of this state. If it exists, its width should be less than 10 keV. We observe a fluctuation of points which might be associated with a narrow 2$^+$ level at excitation energy of 9.29 MeV.\n\nWe noted that the 19$^{th}$ level, 2$^+$, has the adopted \\cite{6} excitation energy in our fit but a different width of 65$\\pm$20 keV. While the two times difference with Ref. \\cite{6} in the widths is marginally exceeds the error bars, the influence on the better fit is evident. The adopted data \\cite{6} for this level are based on the results of a single older work \\cite{22}. The authors \\cite{22} observed a weak $\\gamma$ decay of this state in the presence of a large background. A broader level than in Ref. \\cite{22} was also found in work \\cite{9} as shown in Table 1.\n\nWe characterized the alpha-cluster properties of the states above the alpha particle decay threshold by SF=$\\gamma_\\alpha$=$\\Gamma_\\alpha$ $_{exp}$\/$\\Gamma_\\alpha$ $_{calc}$, where $\\Gamma_\\alpha$ $_{calc}$  is the single alpha particle width calculated in the $\\alpha$-core potential. To normalize SFs we calculated these values for the well-known alpha-cluster states in \\textsuperscript{16}O.\n\nThe Woods-Saxon potential was used to calculate the limit ($\\Gamma_\\alpha$ $_{cal}$) as the widths of single particle states in the potential. First we tried to fit the widths of known alpha-cluster states, 1$^-$ and 4$^+$, in \\textsuperscript{16}O so that $\\gamma_\\alpha$ = $\\Gamma_\\alpha$ $_{exp}$\/$\\Gamma_\\alpha$ $_{cal}$ $\\sim$ 1.0. The real part of the potential was changed to fit the binding energy of the states. The radius of the potential was chosen to be R = r$_0$ $\\times$12$^{1\/3}$; the Coulomb potential was taken in to account as a charge sphere potential with R$_{coul}$ = R.  We made first calculations (1) with r$_0$ = 1.31 fm and the diffuseness a = 0.65 fm, then we set r$_0$ = 1.23 fm (2), and we finally performed the third calculations (3)  with r$_0$ = 1.23 fm and a = 0.6 fm. The results are summarized in Table 2. The $\\gamma_\\alpha$ calculations for the \\textsuperscript{20}Ne states were made with the final (3) parameters  should be compared with SF given by a theory.\n\n\\section{\\label{sec:level4}Theoretical description of the $\\alpha$ cluster states in \\textsuperscript{20}$\\textbf{Ne}$}\n\nCNCIM \\cite{2} is among the latest developments of the classical shell model approaches towards clustering. This model targets a combination of classical configuration interaction techniques with algebraic methods that emerge in the description of clustering.  The ability to construct a fully normalized set of orthogonal cluster channels is at the core of this approach; the overlaps of the shell model states with these channels are associated with spectroscopic factors and compared in Table 1 with the reduced widths obtained earlier. The CNCIM allows us to study clustering features that emerge in models with well-established traditional shell model Hamiltonians. These effective model Hamiltonians are built from fundamental nucleon-nucleon interactions followed by phenomenological adjustments to select observables; thus, they generally describe of a broad scope of experimental data with high accuracy. Apart from using these phenomenological shell model Hamiltonians, our study does not involve any adjustable parameters. In order to fully explore the problem, we considered several different model spaces and corresponding Hamiltonians: the \\textit{sd} model space with USDB interaction \\cite{5}; unrestricted \\textit{p-sd} shell model Hamiltonian \\cite{23}, the same Hamiltonian has been used in Refs. \\cite{2};  WBP Hamiltonian \\cite{24} allowing 0$\\hbar\\omega$, 1$\\hbar\\omega$  and 2$\\hbar\\omega$ excitations in \\textit{p-sd-pf} valence space; and the \\textit{sd-pf} Hamiltonian \\cite{25}. This sequence of Hamiltonians represents and expansions of the valence space from \\textit{sd} to \\textit{p-sd}, to \\textit{p-sd-pf}. All models are in good agreement for the \\textit{sd}-states; the low-lying negative parity states as well as positive 2ph excitations are dominated by the \\textit{p-sd} configurations. Thus, in Table 1 we only include the results from the \\textit{p-sd} Hamiltonian which turned out to be most representative although the following discussion and conclusions are largely based on comparisons. The lowest states associated with significant \\textit{fp} shell component appear at excitation energies above 15 MeV.\n\nShell model calculations for \\textsuperscript{20}Ne with open \\textit{2s-1d (sd)} shells predict well the ground state band. The structure of this band is based on the dominating SU(3) configuration (about 75\\%) with quantum numbers (8,0). The model predicts (Table 1) large SF for all members of the band based on the ground state in \\textsuperscript{20}Ne. The 0$^+$, 2$^+$, and 4$^+$ members of the band are below the $\\alpha$ particle decay threshold and do not have observable $\\alpha$ particle widths. However, large $\\alpha$ cluster SFs for these states provide for a better \\textit{R} matrix fit. While uncertainties for the \\textit{R} matrix amplitudes for these states are large, the fit (Fig. 2 and 3) requires these amplitudes to be close to those of the negative parity states with the known extreme $\\alpha$ cluster structure. The $\\alpha$ particle widths of the highest 6$^+$ and 8$^+$ members of this band are known. There is a long history of attempts and ideas to describe these widths using shell model approaches (see, for instance \\cite{5}). The most calculations predicted large clustering for the band but could not explain the decrease of the reduced width for the 6$^+$ and 8$^+$ members. As one sees in Table 1, the CNCIM calculations are in fine agreement with the experimental data for these states. All members of this band have significant clustering that diminishes at higher energies due to configuration mixing.  The second 0$^+$ state within \\textit{sd} space appears at around 6.7 MeV of excitation (in \\textit{psd} model in Table 1, this is a third 0$^+$ state at 6.9 MeV). This level also has a substantial clustering component and absorbs nearly all 15\\% of the remaining strength of the SU(3) (8,0) component. The following 2$^+_2$ (in experiment and in \\textit{sd} model, but third in \\textit{p-sd} model as discussed in what follows) at about 7.4 MeV of excitation energy being a member of the 0$^+_2$ band can be described in a similar way.\n\nA defrost of the \\textit{1p} shell (which is filled in \\textsuperscript{16}O) results in the doubling of the levels, as it is observed and is shown in Table 1 new levels 0$^+$ and 2$^+$, marked with asterisks, appear. In our calculations, the ordering in energy is reversed for both doublets. Structurally the members of each doublet are very different which allows them to be so close in energy and inhibits configuration mixing and Wigner repulsion. One of the doublet levels (the lower 0$^+$ and 2$^+$ levels) has a large $\\alpha$ cluster SF relative to the first excited state in \\textsuperscript{16}O and much smaller SF relative to the ground state in \\textsuperscript{16}O (pay attention that the theory gives the wrong order for the levels in question, see Table 1). Indeed the predicted difference in the structure is supported by population selectivity in different nuclear reactions. The 6.72 MeV 0$^+$ and 7.42 MeV 2$^+$ are populated much stronger than neighboring 7.20 and 7.83 MeV levels in the \\textsuperscript{16}O(\\textsuperscript{6}Li, d) reaction \\cite{26}. The opposite is the case in the \\textsuperscript{12}C(\\textsuperscript{9}Be, n) or \\textsuperscript{12}C(\\textsuperscript{12}C,$\\alpha$) \\cite{27} reactions. \n\nThe theory gives large single particle spectroscopic factors for the \\textit{sd} states and smaller for the 7.20 and 7.83 MeV states. Indeed, one expects that states in \\textsuperscript{20}Ne with a hole in the \\textit{p1\/2} shell will be weakly excited in the single nucleon transfer, \\textsuperscript{19}F(\\textsuperscript{3}He, d) reaction in accordance with the experimental data \\cite{28}. The experiment \\cite{11,12,28} supports also detailed single particle SF calculations giving SF for the ground state smaller than for the 0$^+$ 6.73 MeV state and much higher SF for the first excited 2$^+$ than for 2$^+$ member of the band based on the 6.73 MeV state.\n\nIn the \\textit{sd} valence space (USDB) the third 0$^+$ state appears only at 11.9 MeV, and it has a relatively small alpha spectroscopic factor; opening of the \\textit{p}-shall in addition to 0$^+$ at 6.27 MeV, leads to 0$^+$ state at 9.7 MeV. However, the predicted state has a low proton spectroscopic factor which is not as well supported by observations. A similar serious discrepancy is observed with a broad 4$^{th}$ 2$^+$ state at around 9 MeV, both \\textit{sd} and \\textit{p-sd} shell models produce candidates but with very low alpha SF.  As evident from our studies the only strong coupling to alpha channels could come from \\textit{fp} shell and higher shells. The lowest two bands saturate the alpha strength within \\textit{sd} configurations, holes in the \\textit{p}-shell do not lead to a significant contribution due to low level of core excitation in the ground state of \\textsuperscript{16}O. While our models predict high excitation energies of states with significant \\textit{fp} components, we can speculate that strong configuration mixing, collective effects such as deformation, and coupling to continuum via super radiance mechanism \\cite{29,30} can enhance admixture needed to reproduce the broad resonances observed. There is a similar problem with $\\alpha$ cluster negative parity states 1$^-$ and 3$^-$, the \\textit{p-sd} Hamiltonian produces an acceptable spectrum but the alpha spectroscopic factors are low (see also \\cite{31}).\n\n\\section{\\label{sec:level5}Conclusions}\n\nIn this work we study $\\alpha$-clustering in \\textsuperscript{20}Ne. This nucleus is a benchmark example of many theoretical techniques targeting clustering in light nuclei. Our \\textit{R}-matrix analysis of TTIK experimental data confirms previously known results and establishes new constraints for the positions and widths of the resonances. We compared our findings with those obtained theoretically using cluster-nucleon configuration interaction approach developed in several previous works \\cite{2} and references therein. There is good overall agreement between theoretically predicted and observed spectra. Our theoretical approach describes very well the ground state band and the band built on the first 0$^+$ state. Allowing cross shell excitations from the \\textit{p}-shell, it was possible to reproduce the band built on the second 0$^+$ state. For these states, all spectroscopic factors for alpha transitions to the ground state of \\textsuperscript{16}O and to the first excited state in \\textsuperscript{16}O as well as proton spectroscopic factors to the ground state of \\textsuperscript{19}F are well reproduced. The situation is not as good when it comes to resonances 1$^-$ 3$^-$ and 4$^{th}$ 0$^+$ and 4$^{th}$ 2$^+$, all of these states are broad and have exceptionally large alpha spectroscopic factors. \n\nIn order to describe strong clustering features these states must include configurations from \\textit{fp} shell and from higher oscillator shells, however Hamiltonians that we explored predict these contributions to be negligible below 15 MeV of excitation. Thus, inability of theoretical models to describe broad states exclusively while working well elsewhere suggests an additional coupling mechanism unaccounted for in the traditional shell model Hamiltonians. The super radiance suggested in Refs. \\cite{29,30} could provide this mechanism. Alternatively, the problem could be associated with relatively unknown cross shell interactions. Therefore, work shows that the experimental study of alpha clustering represents an outstanding tool for exploring cross shell excitations especially those of multi-particle multi-hole nature.\n\n\\begin{acknowledgments}\nThis work was supported by Ministry of Education and Science of the Republic of Kazakhstan (grant number \\#0115\u0420\u041a03029 ``NU-Berkeley\", 2014-2018; grant number \\#0115\u0420\u041a022465, 2015-2017). This material is also based upon work supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Grants No DE-FG02-93ER40773 and No. DE-SC0009883.  G.V.R is also grateful to the Welch Foundation (Grant No. A-1853).\n\\end{acknowledgments}\n\n\\nocite{*}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}