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 ∈ S implies y / x ∉ A . Let ρ ¯ ( A ) = sup S δ ¯ ( S ) and ρ ¯ ( A ) = sup S δ ¯ ( S ) , where the supremum is taken over all A -quotient-free sets S , δ ¯ ( S ) and δ ¯ ( S ) are the upper and lower asymptotic densities of S respectively. Let ρ ( A ) = sup S δ ( S ) , where the supremum is taken over all A -quotient-free sets S such that δ ( S ) exists. In this paper we study the properties of ρ ¯ ( A ) , ρ ¯ ( A ) and ρ ( A ) .