On the number of points in a lattice polytope

TL;DR

This paper proves that for any dimension and modulus, there exists a dilation factor making the lattice point count of a simplicial complex's dilation match its Euler characteristic modulo that number.

Contribution

It establishes a universal dilation factor for simplicial complexes ensuring lattice point counts align with Euler characteristic modulo any given number.

Findings

Existence of a dilation factor for all dimensions and moduli.

Lattice point count equals Euler characteristic mod n after dilation.

Applicable to all simplicial complexes with vertices in ^d.

Abstract

In this article we will show that for every natural $d$ and $n>1$ there exists a natural number $t$ such that for every $d$-dimensional simplicial complex $\mathcal{T}$ with vertices in $\mathbb{Z}^d$, the number of lattice points in the $t^{\mathrm{th}}$ dilate of $\mathcal{T}$ is exactly $\chi(\mathcal{T})$ modulo $n$, where $\chi(\mathcal{T})$ is the Euler characteristic of $\mathcal{T}$.