Density of integral sets with missing differences

TL;DR

This paper investigates the maximum density of integer sets avoiding certain differences, extending known results to cases where the difference set has geometric progression structure.

Contribution

It determines the maximal density for sets avoiding differences in finite geometric progressions, expanding beyond previously solved arithmetic progression cases.

Findings

Maximal density for sets with geometric progression differences determined

Results extend the understanding of difference-avoiding sets beyond arithmetic progressions

Provides new bounds and exact values for specific difference sets

Abstract

Motzkin posed the problem of finding the maximal density $\mu(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In this paper, we determine $\mu(M)$ when $M$ has some other structure. For example, we determine $\mu(M)$ when $M$ is a finite geometric progression.