可配群和范德科普特集合的渐近动力学

Sohail Farhangi, Robin Tucker-Drob
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引用次数: 0

摘要

我们通过证明范德尔科普特集的概念并不依赖于用来定义它的F\o lner序列,回答了伯格森和勒esigne提出的一个问题。我们和罗德里格斯的证明都是通过首先建立可配群的弗斯滕伯格对应原理的反证来进行的。我们和罗德里格斯的证明都是通过首先建立可调和群的弗斯滕伯格对应原理的逆定理来进行的,这涉及到研究雷特序列在群的同余序列上的分布。最后,我们证明了$\mathbb{N}$中许多不涉及F\o lner序列的van derCorput集合的等价特征对于任意可数无限群仍然是等价的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Asymptotic dynamics on amenable groups and van der Corput sets
We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the F\o lner sequence used to define it. This result has been discovered independently by Sa\'ul Rodr\'iguez Mart\'in. Both ours and Rodr\'iguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve F\o lner sequences remain equivalent for arbitrary countably infinite groups.
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