تقرير
Bohr sets in sumsets II: countable abelian groups
| العنوان: | Bohr sets in sumsets II: countable abelian groups |
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| المؤلفون: | Griesmer, John T., Le, Anh N., Lê, Thái Hoàng |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Dynamical Systems, 37A45 (Primary), 11B13, 43A07 (Secondary) |
| الوصف: | We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting $G$ be a countable discrete abelian group and $\phi_1, \phi_2, \phi_3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that (1) If $A \subset G$ has positive upper Banach density and $\phi_1 + \phi_2 + \phi_3 = 0$, then $\phi_1(A) + \phi_2(A) + \phi_3(A)$ contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in $\mathbb{Z}$ and a recent result of the first author. (2) For any partition $G = \bigcup_{i=1}^r A_i$, there exists an $i \in \{1, \ldots, r\}$ such that $\phi_1(A_i) + \phi_2(A_i) - \phi_2(A_i)$ contains a Bohr set. This generalizes a result of the second and third authors from $\mathbb{Z}$ to countable abelian groups. (3) If $B, C \subset G$ have positive upper Banach density and $G = \bigcup_{i=1}^r A_i$ is a partition, $B + C + A_i$ contains a Bohr set for some $i \in \{1, \ldots, r\}$. This is a strengthening of a theorem of Bergelson, Furstenberg, and Weiss. These results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices $[G:\phi_j(G)]$, the upper Banach density of $A$ (in (1)), or the number of sets in the given partition (in (2) and (3)). Comment: 39 pages, 2 figures, incorporating referee's suggestions, to appear in Forum Math. Sigma |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2207.04150 |
| رقم الأكسشن: | edsarx.2207.04150 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |