Lattice points in elliptic paraboloids

TL;DR

This paper investigates the distribution of lattice points in elliptic paraboloids across various dimensions, establishing optimal bounds and extending known results, especially in three dimensions and for simple parabolic regions.

Contribution

It proves the expected optimal exponent for lattice points in elliptic paraboloids in dimensions three and higher, improving upon previous bounds and exploring new aspects in two dimensions.

Findings

Established optimal bounds for lattice points in elliptic paraboloids in $ ext{d} ext{≥} 3$

Improved understanding of lattice point distribution in simple parabolic regions in two dimensions

Extended results to cases where the optimal exponent is conjectural for spheres

Abstract

We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$ because the optimal exponent is conjectural even for the sphere. We also treat some aspects of the case $d=2$, getting for a simple parabolic region an $\Omega$-result that is unknown for the classical circle and divisor problems.