On the density of sumsets, II

Arithmetic quasidensities are a large family of real-valued set functions partially defined on the power set of N, including the asymptotic density, the Banach density and the analytic density. Let B⊆N be a nonempty set covering o(n!) residue classes modulo n! as n→∞ (for example, the primes or the perfect powers). We show that, for each α∈[0,1], there is a set A⊆N such that, for every arithmetic quasidensity μ, both A and the sumset A+B are in the domain of μ and, in addition, μ(A+B)=α. The proof relies on the properties of a little known density first considered by Buck [‘The measure theoretic approach to density’, Amer. J. Math. 68 (1946), 560–580].