Infinite unrestricted sumsets in subsets of abelian groups with large density
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908314072252416 |
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| author | Charamaras, Dimitrios Kousek, Ioannis Mountakis, Andreas Radić, Tristán |
| author_facet | Charamaras, Dimitrios Kousek, Ioannis Mountakis, Andreas Radić, Tristán |
| contents | Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite unrestricted sumsets in subsets of abelian groups with large density Charamaras, Dimitrios Kousek, Ioannis Mountakis, Andreas Radić, Tristán Dynamical Systems Combinatorics 37A15, 05D10 (Primary) 11B13, 11B30 (Secondary) Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense. |
| title | Infinite unrestricted sumsets in subsets of abelian groups with large density |
| topic | Dynamical Systems Combinatorics 37A15, 05D10 (Primary) 11B13, 11B30 (Secondary) |
| url | https://arxiv.org/abs/2504.08649 |