Lattice points inside a random shifted integer polygon

TL;DR

This paper investigates the variance of the number of lattice points inside a randomly shifted integer polygon, providing explicit results for triangles and insights into the distribution of lattice points.

Contribution

It offers new variance estimates for lattice points in shifted polygons and derives an explicit distribution for lattice points in integer triangles.

Findings

Variance of lattice points depends on polygon shape.

Explicit distribution formula for lattice points in integer triangles.

Results enhance understanding of lattice point randomness in polygons.

Abstract

Consider a convex body $C \subset \mathbb{R}^d$. Let $X$ be a random point with uniform distribution in $[0,1]^d$. Consider the value $X_C$ equal to the number of lattice points $\mathbb Z^d$ inside the body $C$ shifted by $X$. It is well known that $\mathbb E X_C = \mathrm{vol}(C)$. The question arises: what can be said about the variance of this random variable? This paper answers this question in the case when $C$ is a polygon with vertices at integer points. Moreover, an explicit distribution of $X_T$ is given for the integer triangle $T$.