论文标题
将Bohr密度与可测量的复发分开
Separating Bohr denseness from measurable recurrence
论文作者
论文摘要
我们证明，有一组整数$ a $具有正面的Banach密度，其差异设置了$ a-a：= \ {a-b：a \} $中的a，b \ in \} $中不包含任何整数的bohr社区，回答了Bergelson，Hegyvári，ruzsa，ruzsa和The Regriptions的问题。在动态系统的语言中，该结果表明，有一组整数$ s $在$ \ mathbb z $的Bohr拓扑中密集，并且不是一组可测量的复发。我们的证明产生以下更强的结果：如果$ \ mathbb z $在$ \ mathbb z $的bohr拓扑中密集\ {0 \} $不是一组可测量的复发。
We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyvári, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $\mathbb Z$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $S\subseteq \mathbb Z$ is dense in the Bohr topology of $\mathbb Z$, then there is a set $S'\subseteq S$ such that $S'$ is dense in the Bohr topology of $\mathbb Z$ and for all $m\in \mathbb Z,$ the set $(S'-m)\setminus \{0\}$ is not a set of measurable recurrence.
