Limit laws in the lattice problem. III. Return to the case of boxes

TL;DR

This paper investigates the statistical behavior of lattice point counts within rectangles as their size grows, revealing convergence properties and limit distributions under various normalizations for different lattice types.

Contribution

It extends previous limit law results to the case of rectangular regions and provides explicit limit distributions and moment calculations for the error in lattice point counting.

Findings

Error normalized by √log(t) converges to a positive constant for random t.

For typical lattices, normalization by log(t) is required for convergence.

When L=Z^2, the error normalized by t converges in distribution, with computed moments.

Abstract

We study the error of the number of points of a lattice $L$ that belong to a rectangle, centred at $0$, whose axes are parallel to the coordinate axes, dilated by a factor $t$ and then translated by a vector $X \in \mathbb{R}^{2}$. When we consider the second order moment of the error relatively to $X \in \mathbb{R}^{2}/L$, one shows that, when $t$ is random and becomes large and when the error is normalized by a quantity which behaves, in the admissible case, as $\sqrt{\log(t)}$, it converges in distribution to an explicit positive constant. In the case of a typical lattice $L$, we show that this result still holds but the normalisation is more important, around $\log(t)$. We also show that when $L=\mathbb{Z}^{2}$, the error, when normalized by $t$, converges in distribution when $t$ is random and becomes large and we compute the moments of the limit distribution.