Limit laws in the lattice problem. V. The case of analytic and stricly convex sets

TL;DR

This paper investigates the asymptotic error behavior in counting lattice points within dilated, strictly convex, analytic sets, extending previous results by analyzing Siegel transforms with random weights and translations.

Contribution

It generalizes earlier work by studying the error distribution of lattice point counts in convex sets, incorporating randomness and translations, and establishing almost sure convergence and moment properties.

Findings

Error normalized by √t converges for random lattices

Siegel transform with random weights converges almost surely

Results hold under translations of convex sets by fixed vectors

Abstract

We study the error of the number of points of a unimodular lattice that fall in a strictly convex and analytic set having the origin and that is dilated by a factor $t$. The aim is to generalize the result of a previous article. We first show that the study of the error, when it is normalized by $\sqrt{t}$, when this parameter tends to infinity and when the considered lattice is random, is reduced to the study of a Siegel transform $\mathcal{S}(f_{t})(L)$ which depends on $t$. Then, we come back to the study of the asymptotic behaviour of a Siegel transform with random weights, $\mathcal{S}(F)(\theta,L)$ where $\theta$ is a second random parameter. Then, we show that this last quantity converges almost surely and we study the existence of moments of its law. Finally, we show that this result is still valid if we translate, after dilation, the strictly convex set of a fixed vector $\alpha \in \mathbb{R}^{2}$.