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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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| _version_ | 1866910453270052864 |
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| author | Leonetti, Paolo Tringali, Salvatore |
| author_facet | Leonetti, Paolo Tringali, Salvatore |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11176 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the density of sumsets, II Leonetti, Paolo Tringali, Salvatore Number Theory 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. |
| title | On the density of sumsets, II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2212.11176 |