Bohr sets in sumsets I: Compact groups
| Autor: | Le, Anh N., Lê, Thái Hoàng |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $G$ be a compact abelian group and $\phi_1, \phi_2, \phi_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $\phi_i$'s, we prove the existence of Bohr sets in the sumset $\phi_1(A) + \phi_2(A) + \phi_3(A)$, 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. Comment: 32 pages, fix minor wordings and typos |
| Databáze: | arXiv |
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