The Gauss Circle Problem and Fourier Quasicrystals

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 $\mathrm{\#}(\mathrm{\Lambda }\cap B_r)=c_0{\text{Vol}}_d\left(B_r\right)+O\left(r^{d-1}\right)$ for any Fourier quasicrystal $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ 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 $\mathrm{\#}(\mathrm{\Lambda }\cap B_r)=c_0{\text{Vol}}_d\left(B_r\right)+O\left(r^{\theta (\mathrm{\Lambda })}\right)$, where $\frac{d-1}{2}<\theta (\mathrm{\Lambda })<d-1$ is some exponent depending on $\mathrm{\Lambda }$. In the special case $d=2$, we also prove lower and upper bounds for the average of the error.