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arXiv · 2109.02378

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

Julien Trevisan
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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 $\mathcal{S}(F)(θ,L)$ where $θ$ 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.

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Julien Trevisan. 2021-09-06. Limit laws in the lattice problem. II. The case of ovals. https://arxiv.org/abs/2109.02378

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