Lattice counting problem

TL;DR

This paper investigates the asymptotic distribution of the error in counting unimodular lattice points within dilated parallelograms, showing convergence to a Cauchy law under certain conditions using ergodic and probabilistic methods.

Contribution

It introduces a novel approach linking ergodic sums over lattice spaces to probabilistic limit laws, extending previous work to the case of parallelograms and providing new asymptotic results.

Findings

Error normalized by log(t) converges to a Cauchy law.

The point process associated with lattice points converges to a Poisson process.

Results suggest similar behavior for higher-dimensional balls with different normalization.

Abstract

We study the error of the number of unimodular lattice points that fall into a dilated and translated parallelogram. By using an article from Skriganov, we see that this error can be compared to an ergodic sum that involves the discrete geodesic flow over the space of unimodular lattices. With the right normalization, we show, by using tools from a previous work of Fayad and Dolgopyat, that a certain point process converges in law towards a Poisson process and deduce that the ergodic sum converges towards a Cauchy centered law when the unimodular lattice is distributed according to the normalized Haar measure. Strong from this experience, we apply the same kind of approach, with more difficulties, to the study of the asymptotic behaviour of the error and show that this error, normalized by $\log(t)$ with $t$ the factor of dilatation of the parallelogram, also converges in law towards a Cauchy centered law when the dilatation parameter tends to infinity and when the lattice and the vector of translation are random. In a next article, we will show that, in the case of a ball in dimension $d$ superior or equal to $2$, the error, normalized by $t^{\frac{d-1}{2}}$ with $t$ the factor of dilatation of the ball, converges in law when $t \rightarrow \infty$ and the limit law admits a moment of order $1$.