Lattice points in a circle for generic unimodular shears

TL;DR

This paper provides an elementary proof that the average squared error in counting lattice points within a circle for all sheared unimodular lattices grows at most proportionally to T log^2(T).

Contribution

It introduces a simple proof showing the mean square of the lattice point remainder is bounded for all shears of a unimodular lattice.

Findings

Mean square of the remainder is bounded by O(T log^2(T))

Elementary proof technique used for the bound

Applicable to all shears of a unimodular lattice

Abstract

Given a unimodular lattice $\Lambda\subseteq \mathbb{R}^2$ consider the counting function $\mathcal{N}_\Lambda(T)$ counting the number of lattice points of norm less than $T$, and the remainder $\mathcal{R}_\Lambda(T)=\mathcal{N}(T)-\pi T^2$. We give an elementary proof that the mean square of the remainder over the set of all shears of a unimodular lattice is bounded by $O(T\log^2(T))$.