The number of points from a random lattice that lie inside a ball

TL;DR

This paper establishes a precise bound on the average number of lattice points within a ball for certain lattice spaces in two and three dimensions, highlighting limitations when considering all lattices.

Contribution

It provides a sharp average bound for lattice point counts in low dimensions and shows such bounds do not extend to the entire lattice space.

Findings

Sharp average bounds in dimensions two and three

Bound does not hold when averaging over all lattices

Highlights differences between restricted and full lattice spaces

Abstract

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.