On multiplicative sidon sets

Fix integers b > a ≥ 1 with g:= gcd(a, b). A set S ⊆ N is {a, b}-multiplicative if ax ≠ by for all x, y ϵ S. For all n, we determine an {a, b}-multiplicative set with maximum cardinality in [n], and conclude that the maximum density of an {a, b}-multiplicative set is b/b+g. For A, B ⊆ N, a set S ⊆ N is {A, B}-multiplicative if for all a ϵ A and b ϵ B and x, y ϵ S, the only solutions to ax = by have a = b and x = y. For 1 < a < b < c and a, b, c coprime, we give a O(1) time algorithm to approximate the maximum density of an {{a}, {b, c}}-multiplicative set to arbitrary given precision.