The Gauss Circle Problem and Fourier Quasicrystals

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.