On the mean square of the remainder for the Euclidean lattice point counting problem

TL;DR

This paper establishes mean square bounds for the error term in counting lattice points within large Euclidean balls, averaged over sheared lattices, advancing understanding of lattice point distribution.

Contribution

It provides new mean square bounds for the lattice point counting remainder when averaged over sheared lattices, a novel approach in this context.

Findings

Mean square bounds for the lattice point counting remainder.

Results hold when averaging over families of sheared lattices.

Improves understanding of lattice point distribution in large Euclidean balls.

Abstract

We give mean square bounds for the remainder in the lattice point counting problem, counting the number of lattice points in a large ball in $\mathbb{R}^d$, when averaged over families of shears of the lattice.