Journal d'Analyse Mathématique ( IF 1 ) Pub Date : 2021-06-29 , DOI: 10.1007/s11854-021-0165-4 Catalin Badea , Sophie Grivaux , Étienne Matheron
We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in ℤ for the Bohr topology, a result originally due to Griesmer.
中文翻译:
整数上的刚性序列、Kazhdan 集和群拓扑
我们研究了三类不同类型的整数序列(或集合)之间的关系,即刚性序列、Kazhdan 序列(或集合)和无效序列。我们证明刚性序列是非 Kazhdan 和无效的,并且所有其他含义都是错误的。特别是,我们通过概率均值证明存在无效且 Kazhdan 的整数序列。此外,使用贝尔类别方法,我们提供了整数序列是刚性序列的一般标准。最后,我们给出了玻尔拓扑在 ℤ 中稠密的刚性序列存在的新证明,这个结果最初是由 Griesmer 提出的。