A Note on the Density of \(M\)-sets in Geometric Sequence

Abstract

For a given set \(M\) of positive integers, a well known problem of Motzkin asks for determining the maximal density \(\mu(M)\) among sets of nonnegative integers in which no two elements differ by an element of \(M\). The problem is completely settled when \(|M| \leq 2\), and some partial results are known for several families of \(M\) for \(|M| \geq 3\),including the case where the elements of \(M\) are in arithmetic progression. We resolve the problem in case of geometric progressions and geometric sequences.