Bohr sets in sumsets I: Compact abelian groups
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| Format: | Preprint |
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2021
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| _version_ | 1866918133696036864 |
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| author | Le, Anh N. Lê, Thái Hoàng |
| author_facet | Le, Anh N. Lê, Thái Hoàng |
| contents | Let $G$ be a compact abelian group and $ϕ_1, ϕ_2, ϕ_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $ϕ_i$'s, we prove the existence of Bohr sets in the sumset $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$, where $A$ is either a set of positive Haar measure, or comes from a finite partition of $G$. The first result generalizes theorems of Bogolyubov and Bergelson-Ruzsa. As a variant of the second result, we show that for any partition $\mathbb{Z} = \bigcup_{i=1}^r A_i$, there exists an $i$ such that $A_i - A_i + sA_i$ contains a Bohr set for any $s \in \mathbb{Z} \setminus \{ 0 \}$. The latter is a step toward an open question of Katznelson and Ruzsa. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_11997 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bohr sets in sumsets I: Compact abelian groups Le, Anh N. Lê, Thái Hoàng Combinatorics Dynamical Systems Number Theory Primary: 05B10, Secondary: 11B13, 37A45 Let $G$ be a compact abelian group and $ϕ_1, ϕ_2, ϕ_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $ϕ_i$'s, we prove the existence of Bohr sets in the sumset $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$, where $A$ is either a set of positive Haar measure, or comes from a finite partition of $G$. The first result generalizes theorems of Bogolyubov and Bergelson-Ruzsa. As a variant of the second result, we show that for any partition $\mathbb{Z} = \bigcup_{i=1}^r A_i$, there exists an $i$ such that $A_i - A_i + sA_i$ contains a Bohr set for any $s \in \mathbb{Z} \setminus \{ 0 \}$. The latter is a step toward an open question of Katznelson and Ruzsa. |
| title | Bohr sets in sumsets I: Compact abelian groups |
| topic | Combinatorics Dynamical Systems Number Theory Primary: 05B10, Secondary: 11B13, 37A45 |
| url | https://arxiv.org/abs/2112.11997 |