On the density of sumsets. II (Q6564326)

The main result of the present paper is stated as follows:\N\N`` ... 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, \(\mu(A+B)=\alpha\)'', where \(B \subseteq \mathbb N\) is a nonempty set covering \(o(n!)\) residue classes modulo \(n!\) as \(n\to \infty\) (it is noted that the set of all primes or the set of all perfect powers are examples of such sets).\N\NThe notion of the arithmetic quasidensity and related notions are explained in detail, several auxiliary properties of upper and lower quasidensities, as well as techniques used in the proof are discussed.\N\NFor Part I see [the authors, Monatsh. Math. 198, No. 3, 565--580 (2022; Zbl 1497.11023)].