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 ${\int }_{{\mathbb{T}}^2}|\mathcal{D}(\Omega ,x,R){|}^{p}dx$ 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 $p\ge 2.$

15 Pages, 5 figures, Revised Version

References in corpus (3)

• $L^p$ and Weak-$L^p$ estimates for the number of integer points in translated domains
• Mixed $L^p(L^2)$ norms of the lattice point discrepancy
• $L^p$ norms of the lattice point discrepancy