Abstract
Abstract
Arithmetic quasidensities are a large family of real-valued set functions partially defined on the power set of
$\mathbb {N}$
, including the asymptotic density, the Banach density and the analytic density. Let
$B \subseteq \mathbb {N}$
be a nonempty set covering
$o(n!)$
residue classes modulo
$n!$
as
$n\to \infty $
(for example, the primes or the perfect powers). We show that, for each
$\alpha \in [0,1]$
, there is a set
$A\subseteq \mathbb {N}$
such that, for every arithmetic quasidensity
$\mu $
, both A and the sumset
$A+B$
are in the domain of
$\mu $
and, in addition,
$\mu (A + B) = \alpha $
. 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].
Publisher
Cambridge University Press (CUP)