Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x, y is an element of S implies y/x is not an element of A. Let (rho) over bar (A) = sup(S) (delta) over bar (S) and (rho) under bar (A) = sup(S) (delta) under bar (S), where the supremum is taken over all A-quotient-free sets S, (delta) over bar (S) and (delta) under bar (S) are the upper and lower asymptotic densities of S respectively. Let rho(A) = sup(S) delta(S), where the supremum is taken over all A-quotient-free sets S such that delta(S) exists. In this paper we study the properties of (rho) over bar (A), (rho) under bar (A) and rho(A). (C) 2009 Elsevier B.V. All rights reserved.