Density of Integral Sets with Missing Differences

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 \(|M| \leq 2\) or \(M\) 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.