Sequences of integers with missing quotients

Let SS be any set of natural numbers, and AA be a given set of rational numbers. We say that SS is an AA-quotient-free set if x,y∈Sx,y∈S implies y/x∉Ay/x∉A. Let ρ¯(A)=supSδ¯(S) and ρ¯(A)=supSδ¯(S), where the supremum is taken over all AA-quotient-free sets SS, δ¯(S) and δ¯(S) are the upper and lower asymptotic densities of SS respectively. Let ρ(A)=supSδ(S)ρ(A)=supSδ(S), where the supremum is taken over all AA-quotient-free sets SS such that δ(S)δ(S) exists. In this paper we study the properties of ρ¯(A), ρ¯(A) and ρ(A)ρ(A).