Limit laws in the lattice problem. IV. The special case of $\mathbb{Z}^{d}$

TL;DR

This paper investigates the asymptotic distribution of the error in counting lattice points within dilated hypercubes in b^d, revealing convergence in law and explicit characteristic functions under specific random dilation and translation models.

Contribution

It provides the first detailed analysis of the limit laws for lattice point counting errors in b^d with random dilations and translations, including explicit characteristic functions.

Findings

Error normalized by t^{d-1} converges in law as T 

Explicit characteristic functions of the limit laws are derived

Results hold for specific random translation models

Abstract

We study the error of the number of points of the lattice $\mathbb{Z}^{d}$ that fall into a dilated and translated hypercube centred around $0$ and whose axis are parallel to the axis of coordinates. We show that if $t$, the factor of dilatation, is distributed according to the probability measure $\frac{1}{T} \rho(\frac{t}{T}) dt$ with $\rho$ being a probability density over $[0,1]$ the error, when normalized by $t^{d-1}$, converges in law when $T \rightarrow \infty$ in the case where the translation is of the form $X=(x,\cdots,x)$ and in the case where the coordinates of $X$ are independent between them, independent from $t$ and distributed according to the uniform law over $[-\frac{1}{2},\frac{1}{2}]$. In both cases, we compute the characteristic function of the limit law.