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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2212.11176 |
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Table of Contents:
- Arithmetic quasi-densities 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, the analytic density, etc. Let $B \subseteq \mathbb{N}$ be a non-empty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty$ (e.g., the primes or the perfect powers). We show that, for each $α\in [0,1]$, there is a set $A\subseteq \mathbb{N}$ such that, for every arithmetic quasi-density $μ$, 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 in 1946.