Multiple ergodic averages and combinatorics

University Of Memphis, Memphis TN

Investigators

Abstract

The research proposed lies in the area of ergodic theory and has several potential applications in combinatorics. In ergodic theory, the investigator plans to carry out an in depth analysis of the limiting behavior of multiple ergodic averages along various integer sequences, including polynomial sequences, sequences related to the prime numbers, Hardy sequences, and random non-lacunary sequences. The tools to be used include recent advances in the theory of characteristic factors and the theory of nilpotent Lie groups. The combinatorial implications are related to problems of finding (and in some cases ``counting'') certain configurations that occur within every subset of the integers with positive density, thus obtaining several far reaching extensions of Szemeredi's theorem on arithmetic progressions. Furthermore, it is plausible that the same configurations occur within the set of primes. The project studies the long term behavior of physical and mathematical systems, evolving over time, whose dynamics is too complicated to be followed in microscopic detail. Ergodic theory introduces mathematical tools for understanding the asymptotic and average behavior of such systems. These tools have also come to play an important role in studying complicated number theoretic and combinatorial structures and the proposed project seeks to bring new light on these areas of research.