Recurrent behavior in rings
This paper discusses the parallel development of certain group theoretic techniques used in the study of group rings and in ergodic theory. Section 1 compares the \(\Delta\)-methods to syndetic and thick sets. Section 2 considers the recurrence capacity of a group action. Section 3 is concerned with the relationship between ergodic theory and the group ring results of \textit{G. Bergman} [Trans. Am. Math. Soc. 157, 459-470 (1971; Zbl 0197.17102)] and \textit{J. E. Roseblade} [Proc. Lond. Math. Soc., III. Ser. 36, 385-447 (1978; Zbl 0391.16008)]. Finally, Section 4 compares almost nilpotent group actions to distal actions. Referees comments. It would appear from this paper that group ring results are now sufficiently powerful to yield algebraic analogues of some basic results in ergodic theory and dynamical systems. In fact these analogs are frequently stronger since they hold in noncommutative situations. Thus there might be some fruitful applications to this direction. On the other hand, it is not clear from this paper that ergodic theory has much to offer the study of group rings other than its colorful notation.
scientific article; zbMATH DE number 4029768