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

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.