Bohr拓扑与某些阿贝尔群的差集
Bohr topology and difference sets for some abelian groups
摘要：
对于固定素数$p$，$\mathbb F_{p}$表示具有$p$元素的域，$\mathbb F_{p}^ω$表示可数直和$\bigoplus_{n=1}^{\infty} \mathbb F_{p}$。将$\mathbb F_{p}^ω$看作可数交换群，构造了一个具有正上Banach密度的集$A\subseteq \mathbb F_{p}^ω$，而差集$A-A:=\{a-b:a,b\in A\}$不包含任何$c\in \mathbb F_{p}^ω$的Bohr邻域.对于$p=2$，我们得到一个更强的结论：$A-A$不包含一个$g+(B-B)$形式的集合，其中$B$是分段联会的。这种建设否定回答了以下问题的几个作者提出的变种：如果$A\subseteq \mathbb Z$有积极的上Banach密度，必须$A-A$包含玻尔邻域的一些$n\in \mathbb Z$？我们还构造了集合$S, A\subseteq \mathbb F_{p}^ω$，使得$S$在$\mathbb F_{p}^ω$的Bohr拓扑中是稠密的，$A$有正的上Banach密度，$A+S$不是分片Bohr.对于$p=2$，我们证明了$S$的每一个平移都是一个拓扑递归集，并且$A+S$不是分片并合的。这些结构回答了作者提出的一个问题的一个变体。
英文摘要：
For a fixed prime $p$, $\mathbb F_{p}$ denotes the field with $p$ elements, and $\mathbb F_{p}^ω$ denotes the countable direct sum $\bigoplus_{n=1}^{\infty} \mathbb F_{p}$. Viewing $\mathbb F_{p}^ω$ as a countable abelian group, we construct a set $A\subseteq \mathbb F_{p}^ω$ having positive upper Banach density while the difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any $c\in \mathbb F_{p}^ω$. For $p=2$ we obtain a stronger conclusion: $A-A$ does not contain a set of the form $g+(B-B)$, where $B$ is piecewise syndetic. This construction answers negatively a variant of the following question asked by several authors: if $A\subseteq \mathbb Z$ has positive upper Banach density, must $A-A$ contain a Bohr neighborhood of some $n\in \mathbb Z$? We also construct sets $S, A\subseteq \mathbb F_{p}^ω$ such that $S$ is dense in the Bohr topology of $\mathbb F_{p}^ω$, $A$ has positive upper Banach density, and $A+S$ is not piecewise Bohr. For $p=2$ we show that every translate of $S$ is a set of topological recurrence and $A+S$ is not piecewise syndetic. These constructions answer a variant of a question asked by the author.
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