The Gauss Circle Problem and Fourier Quasicrystals

TL;DR

This paper extends the classical Gauss circle problem to Fourier quasicrystals, providing improved bounds on lattice point counts within balls and analyzing error terms, especially in two dimensions.

Contribution

It introduces new bounds for lattice point counts in Fourier quasicrystals, refining previous estimates and analyzing average error bounds in two dimensions.

Findings

Improved upper bounds for lattice point counts in Fourier quasicrystals.

Established bounds on the error term exponent depending on the structure.

Derived average bounds for the error in two-dimensional cases.

Abstract

The Gauss circle problem asks for an approximation to the number of lattice points of $\mathbb{Z}^2$ contained in $B_r$, the disk of radius $r$ centered at the origin. Upper, lower, and average bounds have been established for this number-theoretic problem and have been generalized to any lattice in any dimension. We extend this problem to a more general class of structures known as Fourier quasicrystals. Recent work from Alon, Kummer, Kurasov, and Vinzant provides an upper bound $\#(\Lambda \cap B_r) = c_0\textrm{Vol}_d\left(B_r\right)+ O\left(r^{d-1}\right)$ for any Fourier quasicrystal $\Lambda \subset \mathbb{R}^d$ of density $c_0$, where $B_r$ is the $d$-dimensional ball of radius $r$. In this paper, we improve the upper bound for any uniformly discrete Fourier quasicrystal, by showing we can write $\#\left(\Lambda \cap B_r\right) = c_0\textrm{Vol}_d\left(B_r\right) + O\left(r^{\theta(\Lambda)}\right)$, where $\frac{d-1}{2} < \theta(\Lambda) < d-1$ is some exponent depending on $\Lambda$. In the special case $d = 2$, we also prove lower and upper bounds for the average of the error.