Polynomial ergodic theorems and Ramsey theory

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

ABSTRACT: The project concentrates on multiple recurrence and convergence in dynamical systems with a focus on polynomial actions of abelian and nilpotent groups. The problems considered lead to diverse applications of ergodic theory to combinatorics, number theory and algebra which are inaccessible so far by conventional methods. The polynomial Szemeredi and Hales-Jewett theorems proved by the proposers in the recent years have since been extended by the proposers and their colleagues in different directions, each providing a better understanding of the phenomenon of multiple recurrence along polynomials and offering new vistas of research. The directions of study touched upon in this proposal include (but are not limited to) the deeper study of multiple recurrence for nilpotent group actions and the investigation of polynomial multiple recurrence in the framework of IP-systems. Another interesting area considered in this proposal has to do with convergence of ergodic averages naturally appearing in the theory of multiple recurrence and its applications. The results recently obtained by the proposers indicate a dichotomy between the behavior of ergodic averages depending on whether the acting groups have polynomial or exponential growth. The related conjectures formulated in the last section of the proposal are shedding new light on these questions. The theory of multiple recurrence that we focus on in this project uses the ideas from several diverse areas of mathematics and aims to advance our knowledge about the intrinsic properties of dynamical systems which are related to their long range behavior. An example of a highly nontrivial fact stemming from our investigations is the regularity of the behavior of dynamical systems along polynomial time measurements. This fact, in its turn, has strong applications to seemingly distant areas of number theory and combinatorics.