Visible lattice points and the chromatic zeta function of a graph

TL;DR

This paper investigates the probability of certain lattice point configurations in high-dimensional grids and introduces the chromatic zeta function of a graph to generalize these combinatorial problems.

Contribution

It introduces the chromatic zeta function of a graph, extending the analysis of lattice point configurations to more complex graph-induced arrangements.

Findings

Derived probabilities for cycle configurations in high-dimensional lattices

Generalized the problem to arbitrary graphs using the chromatic zeta function

Provided new insights into lattice point distribution related to graph properties

Abstract

We study the probability that a cycle of length k in the lattice [1, n]^s does not contain more lattice points than the k vertices of the cycle. Then we generalize this problem to other configurations induced by a given graph H, introducting the chromatic zeta fuction of a graph.