Moments, tail asymptotics, and universality of the limit law for the shifted-circle lattice-point error term

For a circle of radius R centered at a fixed point alpha in [0,1)^2, let N_alpha(R) count the integer lattice points inside it and let F_alpha(R)=(N_alpha(R)-pi R^2)/sqrt(R). Bleher, Cheng, Dyson and Lebowitz (Commun. Math. Phys. 154, 1993) proved that, when R is averaged over [0,T] with any smooth density, F_alpha(R) converges in distribution to a law with an entire density p_alpha whose tails are stretched-exponentially small with exponent 4 up to epsilon, and they identified the variance as a wild function of the shift. The problem, stated by that work as directions for further research and still open, is to complete the probabilistic description of this limit law and to extend the limit-law framework: (1) prove that the moments of order k>2 of F_alpha converge and equal the moments of p_alpha for every fixed shift (known only for k=2, and for the centered circle up to order 9 by Heath-Brown); (2) determine the sharp two-sided tail asymptotics of p_alpha, closing the epsilon-gap in the exponent 4; (3) extend existence, smoothness, and tail estimates to general strictly convex planar domains for every fixed shift, to dimensions d>=3 for arbitrary (not only Diophantine) centers, and to convex domains with boundary flat points; and (4) prove or refute the analogous limit laws for the oscillatory remainder in Weyl eigenvalue asymptotics of Laplacians on compact manifolds beyond the flat torus (Popov's universality conjecture). The adjacent question of tail universality for Heath-Brown's divisor-problem limit law was resolved by Lau and extended by Lamzouri, and partial extensions exist for shifted balls with Diophantine centers (Bleher-Bourgain), random lattices (Trevisan), and Heisenberg-group norm balls (Gath), but the four questions above remain open.

Background

The Gauss circle problem asks for the discrepancy between $N(R)$, the number of points of the integer lattice $\mathbb{Z}^2$ in a disk of radius $R$, and the area $\pi R^2$; writing $P(R)=N(R)-\pi R^2$, the conjectured truth is $P(R)=O(R^{1/2+o(1)})$ (Hardy's conjecture), while the best proven exponent is still above $0.62$. Because pointwise bounds are so hard, a complementary probabilistic line asks for the statistics of the error: if the radius is randomized — $R$ distributed on $[0,T]$ with a smooth density $\rho(R/T)$, a model for the time average of the fluctuations — does the suitably normalized error have a limit distribution as $T\to\infty$?

Several earlier results shaped this question. Kendall (1948) studied the lattice discrepancy of a randomly placed convex domain (an oval, i.e. a simple closed smooth strictly convex plane curve) and found asymptotic normality when averaging over the domain's position. Heath-Brown (1992) proved that the normalized error $x^{-1/4}\Delta(x)$ in the Dirichlet divisor problem (the discrepancy between the summatory divisor function and its main term) has a limit distribution with a smooth density, that moments up to order 9 exist, and that similar results hold for the error term in the centered circle problem and for the remainder in the mean-square formula for the Riemann zeta function on the critical line. Bleher (1992) proved a general limit-distribution criterion for Besicovitch-almost-periodic functions — functions approximable in mean square on long intervals by finite trigonometric sums — and applied it to the error term of a dilated oval for almost every shift, giving a variance formula.

Bleher, Cheng, Dyson and Lebowitz (BCDL, 1993) then settled the shifted circle for every fixed center $\alpha\in[0,1)^2$. Their starting point is an expansion of the normalized error

$$F_\alpha(R)=\frac{N_\alpha(R)-\pi R^2}{\sqrt R}=\sum_{m\ \mathrm{square\!-\!free}} f_m(\sqrt m\,R;\alpha)+o(1)\ \ \mathrm{(mean\ square)},$$

where $f_m(t;\alpha)=\pi^{-1}\sum_{k\ge1}\frac{r_\alpha(mk^2)}{(mk^2)^{3/4}}\cos(2\pi kt-3\pi/4)$ and $r_\alpha(n)=\sum_{n_1^2+n_2^2=n}e^{2\pi i(n_1\alpha_1+n_2\alpha_2)}$ is a shift-twisted representation number (the Fourier transform of the lattice point measure on the circle of radius $\sqrt n$ evaluated at $\alpha$). Since the square roots of distinct square-free integers (positive integers not divisible by any perfect square greater than $1$) are linearly independent over the rationals, the phases $\sqrt m\,R$ equidistribute as $R$ varies, and BCDL proved that $F_\alpha(R)$ converges in law — independently of the averaging density $\rho$ — to the random series $\eta_\alpha=\sum_m f_m(t_m;\alpha)$ with independent uniform phases $t_m\in[0,1]$. The limit law $v_\alpha$ is absolutely continuous, its density $p_\alpha$ extends to an entire function, and for every $\varepsilon>0$ one has the stretched-exponential tail bounds $p_\alpha(x)\le\exp(-|x|^{4-\varepsilon})$ for large $|x|$ while the tails $P_\alpha(-x)$ and $1-P_\alpha(x)$ of the distribution function are not smaller than $\exp(-x^{4+\varepsilon})$. The variance converges and equals $D_\alpha=(2\pi^2)^{-1}\sum_m|r_\alpha(m)|^2m^{-3/2}$, a continuous but nowhere-analytic function of $\alpha$ with sharp local maxima and infinite one-sided derivatives at every rational point (Bleher-Dyson 1994).

This limit law matters beyond lattice point counting. On a flat two-dimensional torus the eigenvalue counting function of the Laplacian differs from the main term of Weyl's law (the asymptotic $N(\lambda)\sim(\mathrm{area}/4\pi)\lambda$ for eigenvalue counting) by exactly the circle problem error, so BCDL describes the energy-level fluctuations of the simplest integrable quantum system (a system whose classical trajectories are constrained by conserved quantities, as on the flat torus). In physics this is the setting of the Berry-Tabor conjecture, the expectation that quantum spectra of classically integrable systems show Poisson-type rather than random-matrix statistics; the distribution of $F_\alpha$ is the rigorous counterpart of what that literature calls the mode fluctuation distribution. In this context BCDL acknowledged several directions left open: convergence of higher moments; extensions to dimensions $d>2$ and to general strictly convex domains; the effect of boundary flat points (boundary points where the curvature vanishes); analogues for the oscillatory remainder in Weyl asymptotics on compact manifolds beyond the torus; and the sharpness and universality of the tail exponent $4$ — including for the limit laws of Heath-Brown's type.

Problem Statement

Setting. For a fixed center $\alpha\in[0,1)^2$ let $N_\alpha(R)=\#\{n\in\mathbb{Z}^2:\ |n-\alpha|\le R\}$ and $F_\alpha(R)=(N_\alpha(R)-\pi R^2)/\sqrt R$, and let $R$ be random on $[0,T]$ with density $\rho(R/T)$ for a fixed smooth probability density $\rho$ on $[0,1]$ (the limits in question are known to be independent of $\rho$). Bleher-Cheng-Dyson-Lebowitz proved that $F_\alpha$ converges in distribution to a law $v_\alpha$ with an entire density $p_\alpha$ (a probability density on the real line that extends to an entire function on the complex plane), with stretched-exponential tail bounds of exponent $4$ up to $\varepsilon$, and that its variance equals $D_\alpha=(2\pi^2)^{-1}\sum_m|r_\alpha(m)|^2m^{-3/2}$. The problem is to complete the probabilistic description of this limit law and to extend the same limit-law phenomenology across geometries and operators, in the following linked questions stated by the source work as directions for further research:

1. (Higher moments.) For every fixed $\alpha$ and every integer $k>2$, determine whether

$$\lim_{T\to\infty}\frac{1}{T}\int_0^T (F_\alpha(R))^k\,\rho(R/T)\,dR$$

exists, and whether it equals the corresponding moment $\int_{-\infty}^{\infty}x^k\,p_\alpha(x)\,dx$ of the limit density. (For the centered circle $\alpha=0$, moments up to order 9 were proved by Heath-Brown; for general $\alpha$ only $k=2$ is known.)

2. (Sharp tails.) Close the gap between the upper bound $p_\alpha(x)\le\exp(-|x|^{4-\varepsilon})$ and the lower bound $P_\alpha(-x),\,1-P_\alpha(x)\ge\exp(-x^{4+\varepsilon})$ for every $\varepsilon>0$: determine the true large-deviation asymptotics of $p_\alpha$ — in particular whether the tails are of order $\exp(-\Theta(x^4 L(x)))$ for an explicit slowly varying function $L$, as is known for the divisor-problem limit law — including how the tail constants depend on the shift $\alpha$. (The centred circle $\alpha=0$ case is partially covered by Lamzouri; question 2 asks for the general fixed shift.)

3. (General domains and dimensions.) Extend the existence, absolute continuity, entireness, and tail estimates for the limit distribution of the normalized error term to: (a) strictly convex planar domains (ovals) for every fixed shift $\alpha$, beyond the circle and beyond almost-every-shift results; (b) dimensions $d\ge3$, for balls and strictly convex bodies with arbitrary fixed centers, not only centers satisfying a Diophantine condition (a Diophantine center is one whose coordinates are badly approximable by rationals at a polynomial rate — the known $d\ge3$ results cover only such centers); and (c) convex domains whose boundary has flat points (points of vanishing curvature), where the dominant contribution to the error changes and the correct normalization ($\sqrt R$ or a higher power of $R$) is to be determined and the limit distribution established under it — including the interaction between the geometric order of flatness and the arithmetic nature of the tangent slope at the flat points.

4. (Weyl-remainder analogues.) Prove or refute the analogues of these limit laws for the oscillatory remainder in Weyl's eigenvalue asymptotics for Laplacians (more generally, elliptic operators) on compact manifolds other than the flat torus — the closed surfaces with integrable but non-arithmetic geodesic flow being a representative instance rather than the whole target class — as formalized in the universality conjecture linking the circle problem error with the second term of Weyl's formula.

The task is to determine the moment structure and the sharp tail behavior of the shifted-circle limit law, and whether the same limit-law phenomenology extends across the listed domains, dimensions, center-arithmetic classes, and operators.

All four questions together constitute the problem, but each is an independently resolvable sub-problem: settling (or disproving) any one of them under the stated hypotheses resolves that question and leaves the others open, and the three parts (a)-(c) of question 3 are likewise independent of one another.

References

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