$L^p$ norms of the lattice point discrepancy

TL;DR

This paper estimates the $L^p$ norms of the discrepancy between the volume and the count of lattice points in dilated and translated convex bodies with smooth, positively curved boundaries, as the dilation factor grows large.

Contribution

It provides new bounds on the $L^p$ norms of lattice point discrepancy for convex bodies with smooth, positively curved boundaries, extending previous results to a broader setting.

Findings

Derived bounds for $L^p$ norms of discrepancy as $R o \infty$

Extended lattice point discrepancy estimates to convex bodies with smooth, curved boundaries

Analyzed discrepancy behavior over measures supported on positive real axis

Abstract

We estimate the $L^{p}$ norms of the discrepancy between the volume and the number of integer points in $r\Omega-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $\Omega$ in $\mathbb{R}^{d}$ with smooth boundary with strictly positive curvature, \[ \left\{ {\displaystyle\int_{\mathbb R}}{\displaystyle\int_{\mathbb{T}^{d}}}\left\vert \sum_{k\in\mathbb{Z}^{d}}\chi _{r\Omega-x}(k)-r^{d}\left\vert \Omega\right\vert \right\vert ^{p}dxd\mu(r-R) \right\} ^{1/p}, \] where $\mu$ is a Borel measure compactly supported on the positive real axis and $R\to+\infty$.