Arnold's questions on the statistics of period lengths of continued fractions of $\sqrt{Q}$: average order, large values, and Gauss–Kuz'min behavior

For a positive integer $Q$, let $T(Q)$ be the period length of the simple continued fraction of $\sqrt{Q}$ (with $T(Q)=0$ when $Q$ is a square). Arnold's 2009 empirical study of these periods, together with his 2008 companion paper, asked how $T(Q)$ is distributed as $Q$ grows: how frequently the period is odd, how large $T(Q)$ can be relative to $\sqrt{Q}$, what its typical Cesàro-mean order is, and whether the partial quotients of $\sqrt{Q}$ follow the Gauss–Kuz'min statistics known for almost all real numbers. Earlier work has settled several parts of the program: explicit-constant upper bounds $T(Q)\ll\sqrt{Q}\log Q$ predate Arnold's paper; the density of odd periods among eligible squarefree $Q$ is exactly $1-\prod_{j\ge1,\,j\ \mathrm{odd}}(1-2^{-j})\approx0.580577$ by Koymans–Pagano's proof of Stevenhagen's conjecture; Arnold's middle-element parity observations were proved by Li–Ma; and the averaged Gauss–Kuz'min statistics for quadratic irrationals were proved by Ustinov (and, for prime multiples of a fixed quadratic irrational, by Aka). The surviving open core asks for the true Cesàro mean of $T(Q)$ (Arnold's conjectured average order $C\sqrt{x}$), the maximal order of $T(Q)/\sqrt{Q}$ (closing the gap between $\gg\sqrt{Q}/\log\log Q$ and $\ll\sqrt{Q}\log Q$), and the $\sqrt{Q}$-specific Gauss–Kuz'min behavior: pooled convergence to the universal statistics, the growth of the mean period element $\lambda(Q)$, and the structural constraints behind Arnold's entropy paradox.

Background

Every irrational real number has an infinite simple continued fraction $[a_0;a_1,a_2,\dots]$ with integer partial quotients $a_i\ge1$ for $i\ge1$. By a classical theorem of Lagrange, the expansion is eventually periodic exactly for quadratic irrationalities. For a positive integer $Q$ that is not a square, write

$$\sqrt{Q}=[a_0;\overline{a_1,\dots,a_{T(Q)}}],$$

where the overbar marks the minimal repeating block; $T(Q)$ is the period length, and one sets $T(Q)=0$ for perfect squares. The period has a palindromic structure (a theorem of Galois): $a_T=2a_0$ and $a_i=a_{T-i}$ for $1\le i\le T-1$. In the standard expansion algorithm the complete quotients (the quadratic irrationalities remaining after each step) take the form $(\sqrt{Q}+m_k)/v_k$ with integers $m_k,v_k$ satisfying $v_k\mid Q-m_k^2$ and $0<m_k<\sqrt{Q}$; since each pair $(m_k,v_k)$ determines the next step, $T(Q)$ is bounded by the number of admissible pairs. Hickerson exploited this through the function $g(d)$ counting the pairs $(m,q)$ with $0<m<\sqrt{d}$, $\sqrt{d}-m<q<\sqrt{d}+m$, and $q\mid d-m^2$, for which $T(d)\le g(d)$. The parity of $T(Q)$ is arithmetic rather than statistical: $T(Q)$ is odd if and only if the negative Pell equation $x^2-Qy^2=-1$ is solvable in integers. Solvability forces $Q$ to be a sum of two coprime squares, $Q=a^2+b^2$ with $\gcd(a,b)=1$; call such $Q$ eligible. Arnold's tables for $Q<200$ record eligible $Q$ with even period, such as $T(34)=4$, $T(146)=2$, $T(194)=4$ (and the square $Q=169=13^2$, for which $T=0$); he noted that the observed prevalence of even period lengths over odd ones, and the absence of odd periods for $Q<200$ divisible by $4$, have no explanation. For eligible squarefree $d$, Stevenhagen conjectured, and Koymans–Pagano recently proved, that the density of solvable negative Pell equations equals $1-\prod_{j\ge1,\,j\ \mathrm{odd}}(1-2^{-j})=0.58057\ldots$. The size of $T(Q)$ is tied to the fundamental unit $\varepsilon_Q>1$ of the order $\mathbb{Z}[\sqrt{Q}]=\{a+b\sqrt{Q}:a,b\in\mathbb{Z}\}$ (the smallest real number larger than $1$ invertible in this ring; its logarithm $\log\varepsilon_Q$ is the regulator of the real quadratic field $\mathbb{Q}(\sqrt{Q})$, the standard measure of the size of its unit group): Cohn proved $T(d)>\log\varepsilon_d/\log d$, while convergent denominators grow at least like Fibonacci numbers, so $T(d)$ and $\log\varepsilon_d$ agree up to a factor $\log d$. Maximal-order questions about $T$ therefore contain, up to a logarithmic factor, the classical problem of how large regulators of real quadratic fields can be. For almost all real numbers, the Gauss–Kuz'min theorem asserts that the frequency of the partial quotient $a$ tends to $g_a=\frac{1}{\log 2}\log\left(1+\frac{1}{a(a+2)}\right)$. Arnold's program asks to what extent the periods of $\sqrt{Q}$, pooled over $Q$, display the same statistics, and to what extent they are constrained. He observed empirically that $T(Q)\le C\sqrt{Q}\log Q$; that the mean period element $\lambda(Q)=\Sigma(Q)/T(Q)$, where $\Sigma(Q)$ is the sum of the period's elements, grows roughly like $Q^{0.24}$ (the exponent is observed for the equivalent family of roots of $x^2+x=Q$, i.e. of numbers $(\sqrt{1+4Q}-1)/2$); and he noted an entropy paradox: if the roughly $X$ integers $Q\le X$ all carried periods statistically indistinguishable from Gauss–Kuz'min-random words, then, since the Gauss map $u\mapsto\{1/u\}$ generating continued fractions has entropy $h$ and admits only about $e^{hT}$ words of length $T$, typical periods could not exceed length $O(\log X)$, yet the mean period appears to grow of order about $\sqrt{X}$. His 2008–2009 papers posed these observations as problems for theory, and they anchor the questions recorded below.

Problem Statement

Let $T(Q)$ denote the period length of the simple continued fraction of $\sqrt{Q}$ (with $T(Q)=0$ for perfect squares), let $\Sigma(Q)$ denote the sum of the elements of one period, and let $\lambda(Q)=\Sigma(Q)/T(Q)$ be the mean period element. The common objective, in Arnold's formulation, is to replace the empirical regularities observed for the periods of $\sqrt{Q}$ by theorems. The following parts of Arnold's program remain open. Parts (i), (ii) and (iii) are independent: each may be settled on its own, independently of the others; within (iii), the sub-parts (a), (b) and (c) are jointly required. (i) Average order. Determine the order of the Cesàro mean $A(x)=\frac{1}{x}\sum_{Q\le x}T(Q)$. This upper bound $A(x)\le c_1\sqrt{x}$ with explicit $c_1$ is known, and Arnold conjectured that the mean period over $Q\le x$ grows proportionally to $\sqrt{x}$ (in his notation $T\sim C\sqrt{R}$ for the mean over the disc of radius $R$ in the $(p,q)$-plane of the family of roots of $x^2+px+q=0$; equivalently, for the $\sqrt{Q}$ family, that $\sum_{Q\le x}T(Q)$ has order $x^{3/2}$ — his conjecture $\alpha=3/2$, for which he recorded that there is as yet no proof). Prove or disprove this conjectured average order, that is, whether $A(x)\asymp\sqrt{x}$ as $x\to\infty$. (ii) Large values and maximal order. The best published explicit family (Cohn's $Q=2^{2k+1}$, whose fundamental unit is $\varepsilon=(1+\sqrt{2})^{2^k}$, so that $\log\varepsilon\asymp\sqrt{Q}$) yields only $T(Q)\gg\sqrt{Q}/\log\log Q$, while the known upper bound is $T(Q)\le\frac{7}{2\pi^2}\sqrt{Q}\log Q+O(\sqrt{Q}\log\log Q)$; a suggestion of Lehmer recorded by Cohn is that values near $\frac{1.8}{\pi^2}\sqrt{Q}\log Q$ occur infinitely often. Determine $\limsup_{Q\to\infty}T(Q)/(\sqrt{Q}\log Q)$, or at least whether $\limsup T(Q)/\sqrt{Q}=\infty$. Note that proving $T(Q)=o(\sqrt{Q}\log Q)$ would imply $\log\varepsilon_Q=o(\sqrt{Q}\log Q)$, a regulator bound long sought in vain. (iii) Gauss–Kuz'min behavior specific to $\sqrt{Q}$. Averaged Gauss–Kuz'min statistics are known for reduced quadratic irrationals ordered by regulator (Ustinov), which solved Arnold's problem in that ensemble, and for the periods of $px$ along large sets of primes $p$ for a fixed quadratic irrational $x$ (Aka). For the pooled family of square roots $\sqrt{Q}$ with $Q\le X$: (a) prove that the frequency of each fixed partial quotient $a$ among all period elements tends to the Gauss–Kuz'min value $g_a$ as $X\to\infty$; (b) prove that the mean period element $\lambda(Q)$ tends to infinity along the Cesàro average — Arnold's observed growth of order about $Q^{0.24}$ (for the equivalent family of roots of $x^2+x=Q$) is unproved in general — and determine the order of the Cesàro mean $\frac1x\sum_{Q\le x}\lambda(Q)$; (c) resolve Arnold's entropy paradox by identifying the structural constraint (palindromic or otherwise) that distinguishes the periods of $\sqrt{Q}$ from Gauss–Kuz'min-random words of the same length: for each fixed partial quotient $a$, set $f_a(L)=\lim_{X\to\infty}$ (frequency of $a$ among the period elements of the integers $Q\le X$ with $T(Q)=L$), and prove or disprove the target proposition that $f_a(L)$ does not tend to the Gauss–Kuz'min value $g_a$ as $L\to\infty$, that is, that the convergence to the universal statistics in (a) cannot be uniform in the word length.

References

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