Asymptotic growth law for the averaged period length of the continued fractions of the eigenvalues of integer 2x2 matrices in growing 4-balls (Arnold's 2008 problem)

For a 2x2 integer matrix with real irrational eigenvalues, the eigenvalues are quadratic irrationals and their continued fractions are periodic. Arnold (2008) studied the average period length of these continued fractions over all integer matrices in the Euclidean 4-ball of entries with squared radius M, as M tends to infinity, tabulated it for M <= 16, and recorded in his public problem list the empirical growth law Tb(R) = CR for the mean period length in the ball radius R = sqrt(M) - that is, an average of order Csqrt(M) - in agreement with a nonrigorous heuristic chain, based on the conjectural relation between the period and the square root of the discriminant, predicting that the average grows like csqrt(M). He also proved exact period values for determinant-one families, that the SL(2,Z)-average of the period length tends to 2, and upper bounds for the counts of trace-restricted SL(2,Z) matrices in the ball. The problem left open is to prove or disprove the recorded asymptotic growth law of this average and the underlying averaged heuristic T(p,q) ~ c2*sqrt(p^2 - 4q) with explicit constants and error terms. Later work (Lerner 2010; Aka and Shapira 2018; Battistoni, Grenié and Molteni 2024; Korolev 2024) resolves neighboring questions - weak Gauss-Kuzmin statistics, statistics in a fixed quadratic field, and upper bounds for moments of periods of square roots - but the averaged growth law remains unproved in every relevant ensemble, and no later computation of the 4-ball average at larger M is known. The sharp asymptotics of the trace-restricted counting functions of SL(2,Z) matrices in the ball is a related but separate open problem of Arnold (Problem 14 of his ICTP 2007-2008 list).

Background

Every real number has a regular continued fraction expansion $x=[a_0;a_1,a_2,\ldots]$ with integer partial quotients $a_i$; by Lagrange's theorem the expansion of a quadratic irrational (an irrational root of a quadratic equation with rational coefficients) is eventually periodic, and conversely. Write $T$ for the length of the period. For the quadratic equation $x^2+px+q=0$ with integer coefficients and positive non-square discriminant $\Delta=p^2-4q$, the larger root is a quadratic irrational whose period length is denoted $T(p,q)$. The size of $T(p,q)$ is governed by the arithmetic of the real quadratic field $\mathbb{Q}(\sqrt{\Delta})$: the period is essentially the length of the cycle of reduced binary quadratic forms equivalent to the given one, or equivalently (up to constant factors) a logarithm of the fundamental unit - the smallest solution greater than 1 of the associated Pell equation $u^2-\Delta v^2=4$ - so sharp average statements about $T$ couple to the notoriously difficult statistics of regulators and class numbers of real quadratic fields (the regulator being, up to a logarithm, the size of the fundamental unit, and the class number counting the ideal classes of the field).

The eigenvalues of an integer matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ are the roots of $x^2+px+q=0$ with $p=-(a+d)$ and $q=\det A=ad-bc$, so whenever the discriminant $\Delta=p^2-4q=(a-d)^2+4bc$ is positive and not a perfect square, the eigenvalues are real irrational and their continued fractions are periodic. Arnold studied the statistics of these periods when $A$ ranges over the integer matrices in the 4-ball $a^2+b^2+c^2+d^2\le M$ (the Euclidean ball of squared radius $M$ in the four-dimensional space of entries), with $M\to\infty$, and posed parallel problems for the disk $p^2+q^2\le R^2$ of coefficients and for the subgroup $\mathrm{SL}(2,\mathbb{Z})$ of integer matrices with determinant $1$. For a uniformly random real number the partial quotients follow the classical Gauss-Kuzmin distribution; Arnold asked how the statistics of partial quotients and of period lengths over these quadratic-irrationality ensembles deviate from that benchmark.

In the source paper, Arnold proved: the exact values $T(\pm2,1)=0$, $T(\pm3,1)=1$ and $T(p,1)=2$ for $|p|\ge4$; the upper bound $Q(M)\le K\sqrt M\ln M$ for the number $Q(M)$ of $\mathrm{SL}(2,\mathbb{Z})$ matrices of trace $2$ in the 4-ball; the lower bound $G(M)\ge LM$ for the number $G(M)$ of $\mathrm{SL}(2,\mathbb{Z})$ matrices in the ball with real eigenvalues; the uniform bound $Q_s(M)\le c''M^{5/6}$ for the number $Q_s(M)$ of $\mathrm{SL}(2,\mathbb{Z})$ matrices of any fixed trace $s$ in the ball; and, combining these, that the average period length over the $\mathrm{SL}(2,\mathbb{Z})$ part of the ball tends to the finite limit $2$. He tabulated, for $M\le16$, the total period sum $\Sigma(M)$, the count $Z(M)$ of matrices with real eigenvalues, and the sample mean $\hat\tau(M)=\Sigma(M)/Z(M)$; his public problem-list record of the same ensemble describes the mean period length as growing approximately linearly in the ball radius $R$ ($T_b(R)\approx CR$, i.e. $\hat\tau(M)\approx C\sqrt M$ in the squared-radius variable), and, for $M\le70$, related $\mathrm{SL}(2,\mathbb{Z})$ statistics.

On the heuristic that the period of the larger root of $x^2+px+q=0$ grows on average like the square root of the discriminant, $T(p,q)\sim c_2\sqrt{p^2-4q}$, the paper argues a nonrigorous chain: on each sphere $a^2+b^2+c^2+d^2=N$ the discriminant is typically of order $N$ while the number of matrices with real eigenvalues on the sphere is of order $N$, whence the predicted asymptotics $\Sigma(M)\sim c_6M^{5/2}$, $Z(M)\sim c_7M^2$ and $\hat\tau(M)\sim c_8\sqrt M$, in agreement with the empirically recorded growth of the sample mean linear in the ball radius. (In the coefficient-disk ensemble the same heuristic instead supports Arnold's conjecture that the mean period grows linearly, $\bar T(R)\sim cR$ with $c\approx0.15$ reported empirically; and for $x^2-d=0$ it underlies the classical open problem of the average size of the period $T(d)$ of $\sqrt d$, for which only pointwise bounds of order $\sqrt d\log d$ - improved to $\sqrt d\log\log d$ under the generalized Riemann hypothesis - and upper bounds for the mean are known.) The open target is the rigorous status of this growth law and of the heuristic behind it.

Problem Statement

Let $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ range over all $2\times2$ integer matrices in the 4-ball $a^2+b^2+c^2+d^2\le M$, and for each matrix with real eigenvalues let $T$ be the length of the period of the continued fraction of the larger eigenvalue (equivalently, of the larger root of $x^2+px+q=0$ with $p=-(a+d)$ and $q=ad-bc$), adopting Arnold's convention that a matrix with rational eigenvalues - that is, with $\Delta=p^2-4q$ a perfect square - contributes $T=0$; such matrices are asymptotically negligible in the ball, as Arnold proved. Let $\Sigma(M)$ be the sum of these period lengths over the ball, $Z(M)$ the number of matrices in the ball with real eigenvalues, and $\hat\tau(M)=\Sigma(M)/Z(M)$ their arithmetic mean (equivalently, the mean over the lattice points of the ball with $\Delta>0$ under the convention $T=0$ for rational roots).

The problem, as left open by Arnold (2008), is to determine the asymptotic statistics of the averaged period length as $M\to\infty$ (equivalently, as the ball radius $R=\sqrt M\to\infty$), specifically:

1. Growth law of the average: prove or disprove that $\hat\tau(M)\sim C\sqrt M$ for some constant $C>0$ - equivalently, that the mean period length grows linearly in the ball radius, $\hat\tau(R)\sim CR$, as recorded empirically by Arnold - or determine the true asymptotic behavior of $\hat\tau(M)$ if this law fails.

2. Rigorous averaged heuristic: prove or disprove the averaged form of the heuristic $T(p,q)\sim c_2\sqrt{p^2-4q}$, namely that the ratio of the mean of $T(p,q)$ to the mean of $\sqrt{p^2-4q}$ over the integer lattice points of the 4-ball $B(M)=\{A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathbb{Z}^{2\times2}:a^2+b^2+c^2+d^2\le M\}$ with $\Delta=p^2-4q>0$ tends to $c_2$ as $M\to\infty$ - explicitly, $\big(\sum_{A\in B(M),\,\Delta>0}T(A)\big)\big/\big(\sum_{A\in B(M),\,\Delta>0}\sqrt{p^2-4q}\big)\to c_2$, where $T(A)=T(p,q)$, $p=-(a+d)$, $q=ad-bc$ and $\Delta=(a-d)^2+4bc$ - under the convention $T=0$ for rational roots, with explicit constants and error terms; this is the load-bearing unproven ingredient of the $\sqrt M$ prediction.

Both deliverables share one objective - the asymptotic law of the averaged period length - the averaged heuristic being the input that produces the predicted growth law. Item 2 is the stronger claim, and proving it discharges item 1: the averaged heuristic, combined with the counting asymptotics of the ball, yields the growth law $\hat\tau(M)\sim C\sqrt M$. Item 1 remains an independent requirement only if the heuristic is disproved, in which case the true asymptotic behavior of $\hat\tau(M)$ must still be determined.

The sharp asymptotics and optimal constants for the counting functions $Q_s(M)$ and $Q(M)$ of trace-restricted $\mathrm{SL}(2,\mathbb{Z})$ matrices in the ball - improving the proved upper bounds $Q(M)\le K\sqrt M\ln M$ and $Q_s(M)\le c''M^{5/6}$ - form a related but separate open problem of Arnold (Problem 14 of his ICTP 2007-2008 list) and are not part of the present problem.

References

1. Arnold, V.I. (2008). Statistics of the period lengths of the continued fractions for the eigenvalues of the integer matrices of order two. Functional Analysis and Other Mathematics 2, 15-26. DOI: 10.1007/s11853-008-0013-4. URL: https://doi.org/10.1007/s11853-008-0013-4
2. Arnol'd, V.I. (2008). Statistics of the periods of continued fractions for quadratic irrationals. Izvestiya: Mathematics 72(1), 1-34. DOI: 10.1070/im2008v072n01abeh002389. URL: https://doi.org/10.1070/im2008v072n01abeh002389
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10. Battistoni, F., Grenie, L., Molteni, G. (2024). The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$. Mathematika 70(4). DOI: 10.1112/mtk.12273. URL: https://arxiv.org/abs/2401.13118
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14. Řada, H., Starosta, Š. (2020). Bounds on the period of the continued fraction after a Möbius transformation. Journal of Number Theory 212, 122-172. DOI: 10.1016/j.jnt.2019.10.027. URL: https://arxiv.org/abs/1901.01383