On integer points inside a randomly shifted polyhedron

TL;DR

This paper investigates the distribution of the number of lattice points inside a randomly shifted polyhedron, extending classical volume-based expectations to probabilistic distribution analysis.

Contribution

It introduces a probabilistic framework for analyzing lattice points in randomly shifted polyhedra with integer vertices, providing new insights beyond classical volume expectations.

Findings

Derived distributional properties of lattice points in shifted polyhedra

Established bounds and probabilistic estimates for lattice point counts

Extended classical volume expectation results to a probabilistic setting

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.