Wednesday, January 22, 2020 - 3:00pm to 4:00pm
Title: Multiple ergodic averages for appropriate sequences and applications
Abstract: In 1977, Furstenberg provided a purely ergodic theoretical proof of Szemerédi’s theorem (i.e., every subset of natural numbers with positive upper density contains arbitrarily long arithmetic progressions) through the study of the behavior of multiple averages for linear iterates. Polynomial extensions of the results from Furstenberg’s study came later by Bergelson (1987) and Bergelson-Leibman (1996), who also conjectured that, under certain assumptions, multiple averages with polynomial iterates always have a limit. Intermediate results to this conjecture were proved by Frantzikinakis-Kra, Tao, Host-Kra, Chu-Frantzikinakis-Host and others. Walsh (2012) was the one to prove a more general statement, answering the full conjecture of Bergelson and Leibman in the positive. However, Walsh’s result provided no explicit expression-information on the limit of the averages of interest. The knowledge of such a limit, even in special cases, led to important applications in dynamical systems, combinatorics and number theory. Extensions to other, more exotic, classes of functions such as the tempered and Hardy field ones were studied recently as well (Bergelson-Knutson, 2009 and Frantzikinakis, 2010-2015 respectively). In this talk I will present some recent developments in the study of the limiting behavior of multiple averages for appropriate, integer-valued, sequences. This work is the culmination of both individual and collaborative work (with S. Donoso, D. Karageorgos and W. Sun).