Density of integral sets with missing differences

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 $\mathrm{\mid }M\mathrm{\mid }\le 2$ 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.