Higher Moments for Lattice Point Discrepancy of Convex Domains and Annuli

Abstract

Given a domain $\Omega \subseteq \mathbb{R}^2$, let $\mathcal{D}(\Omega,x,R)$ be the number of lattice points from $\mathbb{Z}^2$ in $R\Omega-x$, for $R \ge 1$ and $x\in \mathbb{T}^2$, minus the area of $R\Omega$: $$\mathcal{D}(\Omega,x,R) = \# \{ (j,k) \in \mathbb{Z}^2 :(j-x_1,k-x_2) \in R\Omega \} - R^2|\Omega|.$$ We call $\int_{\mathbb{T}^2}|\mathcal{D}(\Omega,x,R)|^pdx$ the $p$-th moment of the discrepancy function $\mathcal{D}$. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of $\mathcal{D}$ is bounded by $\mathcal{O}(R^2\log R)$, 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 $p\ge 2.$