Sets of Integers with Missing Differences

This paper deals with the problem of finding the maximal density, μ(M), of sets of integers in which differences given by a set M do not occur. The problem is solved for the case where the elements of M are in arithmetic progression. Besides finding lower bounds for most members of the general three element set M, μ(M) has been found for most members of the family {i, j, k}, where i/d≡j/d (mod 2) and gcd(i, j)=d. For i/d≢j/d (mod 2) and gcd(i, j)=d it is conjectured that the lower bound found is the best possible. A lower bound is given for μ(M) for the set M={i, j, i+j} and μ(M) for certain infinite families of four element set M have been found.