On lattice points in large convex bodies

TL;DR

This paper improves the estimate of the error term in counting lattice points within large smooth convex bodies with nonzero Gaussian curvature in three or more dimensions.

Contribution

It provides a new, sharper bound on the lattice point remainder for smooth convex bodies with nonzero Gaussian curvature, advancing previous results.

Findings

Enhanced estimate of lattice point remainder in convex bodies

Applicable to bodies in dimensions three and higher

Improved bounds over previous best results

Abstract

We consider a compact convex body $\mathcal{B}$ in $\mathbb{R}^d$ $(d\geqslant 3)$ with smooth boundary and nonzero Gaussian curvature and prove a new estimate of $P_{\mathcal{B}}(t)$, the remainder in the lattice point problem, which improves previously known best result.