Limit laws in the lattice problem. II. The case of ovals

TL;DR

This paper investigates the asymptotic distribution of lattice point counting errors within dilated ovals, revealing convergence properties of associated Siegel transforms and their moments in a probabilistic setting.

Contribution

It introduces a novel analysis of lattice point errors for ovals, linking the problem to Siegel transforms and establishing convergence in law and almost sure results.

Findings

Normalized error converges in law to a modified Siegel transform with random weights.

The modified Siegel transform converges almost surely as the dilation parameter grows.

The paper studies the existence of moments of the limiting distribution.

Abstract

We study the error of the number of unimodular lattice points that fall into a dilated and centred ellipse around $0$. We first show that the study of the error, when the error is normalized by $\sqrt{t}$ with $t$ the parameter of dilatation of the ellipse, when $t$ tends to infinity and when the lattice is random, is reduced to the study of a Siegel transform $\mathcal{S}(f_{t})(L)$ that depends on $t$. Then, by making $t \rightarrow \infty$, we see that $\mathcal{S}(f_{t})$ converges in law towards a modified Siegel transform with random weights $\mathcal{S}(F)(\theta,L)$ where $\theta$ is a second random parameter. Finally, we show that this last quantity converges almost surely and we study the existence of the moments of its law.