Sumsets with Prescribed Lower and Upper Asymptotic Densities II

We show that for natural numbers h, k, such that k ≥ 2, and for real numbers α₁, α₂, such that 0 <α₁ ≤ α₂ < 1, there exists a subset A of ℕ = {0, 1, 2,... }, such that for all finite subsets B of ℕ such that max B − min B = h, the lower and upper asymptotic densities of kA + B are α₁ and α₂, respectively.