The Gauss Circle Problem and Fourier Quasicrystals

The Gauss circle problem asks for an approximation to the number of lattice points of Z² contained in Bᵣ, 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 #(Λ ∩ Bᵣ) = c₀Vold (Bᵣ )+ O (rᵈ⁻¹ ) for any Fourier quasicrystal Λ ⊂ Rᵈ of density c₀, where Bᵣ is the d-dimensional ball of radius r. In this paper, we improve the upper bound for any Fourier quasicrystal by showing we can write # (Λ ∩ Bᵣ ) = c₀Vold (Bᵣ ) + O (rθ (Λ ) ), where d-1/2 < θ (Λ ) < d-1 is some exponent depending only on the dimension d and the growth rate of the spectrum S of Λ. In the special case $d = 2$, we also prove lower and upper bounds for the average of the error.