Higher Moments for Lattice Point Discrepancy of Convex Domains and Annuli

TL;DR

This paper provides a simplified proof of Huxley's 2014 result on the fourth moment of lattice point discrepancy for convex domains and extends the analysis to higher moments for annuli with fixed thickness, revealing new bounds.

Contribution

It offers a straightforward proof of existing fourth-moment bounds and introduces novel estimates for higher moments of lattice point discrepancy in annuli.

Findings

Simplified proof of Huxley's 2014 fourth-moment bound.

New estimates for higher moments of discrepancy in annuli.

Results applicable for fixed annuli thickness and moments p ≥ 2.

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 $p\ge 2.$