论文标题
BOHR集合II：可数的Abelian群体
Bohr sets in sumsets II: countable abelian groups
论文作者
论文摘要
我们证明了三个以三倍总和集中存在的BOHR集合的结果。更准确地说，让$ g $是一个可数的离散的阿贝利安组，$ ϕ_1，ϕ_2，ϕ_3：g \ g $ to g $是通勤的内态图像具有有限索引的内态，我们表明 （1）如果$ a \ subset g $具有正面的BANACH密度，并且$ ϕ_1 + ϕ_2 + ϕ_3 = 0 $，则$ ϕ_1（a） + ϕ_2（a） + ϕ_3（a）$包含bohr集。这将Bergelson和Ruzsa的定理概括为$ \ Mathbb {Z} $，以及第一作者的最新结果。 （2）对于任何分区，$ g = \ bigCup_ {i = 1}^r a_i $，存在$ i \ in \ {1，\ ldots，r \} $，因此$ ϕ_1（a_i） + ϕ_2（a_i） - ϕ_2（a_i） - ϕ_2（a_i）$包含一个BOHR集。这将第二和第三作者的结果从$ \ mathbb {z} $带到了可数的阿贝利安组。 （3）如果$ b，c \ subset g $具有正面的banach密度，而$ g = \ bigcup_ {i = 1}^r a_i $是一个分区，则$ b + c + a_i $包含bohr set，用于某些$ i \ in \ in \ in \ {1，\ ldots，rdots，r \} $。这是伯格森，弗斯滕伯格和魏斯定理的加强。 从某种意义上说，这些结果是定量的，因为获得的BOHR集的半径和等级仅取决于索引$ [g：ϕ_j（g）$，$ a $ a $（（1）中的Banach密度）或给定分区中的集合数（在（2）和（3）中）。
We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, letting $G$ be a countable discrete abelian group and $ϕ_1, ϕ_2, ϕ_3: G \to G$ be commuting endomorphisms whose images have finite indices, we show that (1) If $A \subset G$ has positive upper Banach density and $ϕ_1 + ϕ_2 + ϕ_3 = 0$, then $ϕ_1(A) + ϕ_2(A) + ϕ_3(A)$ 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 $ϕ_1(A_i) + ϕ_2(A_i) - ϕ_2(A_i)$ 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 $[G:ϕ_j(G)]$, the upper Banach density of $A$ (in (1)), or the number of sets in the given partition (in (2) and (3)).
