On integer points inside a randomly shifted polyhedron

Abstract

Consider a convex body $C\subset {\mathbb{R}}^d$. Let $X$ be a random point with uniform distribution in $[0,1{]}^d$. Define $X_C$ as the number of lattice points in ${\mathbb{Z}}^d$ inside the translated body $C+X$. It is well known that $\mathbb{E}{X}_C=\mathrm{vol}(C)$. A natural question arises: What can be said about the distribution of $X_C$ in general? In this work, we study this question when $C$ is a polyhedron with vertices at integer points.