Higher moments for lattice point discrepancy of convex domains and annuli

Abstract

TextGiven a domain Ω⊆R2, let D(Ω,x,R) be the number of lattice points from Z2 in RΩ−x, for R≥1 and x:=(x1,x2)∈T2, minus the area of RΩ:D(Ω,x,R)=#{(j,k)∈Z2:(j−x1,k−x2)∈RΩ}−R2|Ω|. We call ∫T2|D(Ω,x,R)|pdx the p-th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O(R2log⁡R), and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions.In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p-th moments of lattice point discrepancy of annuli of radius R, and any fixed thickness 0<t<1 for p≥2. VideoFor a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q.