Variable polynomials and joint ergodicity for functions of polynomial growth.

RESUMEN: The ergodic theoretical proof of Szemerédi’s theorem on arithmetic progressions by Furstenberg, in 1977, led to a thorough study of multiple ergodic averages; which in turn gave numerous far-reaching extensions of Szemerédi’s result. More specifically, we have polynomial (Bergelson-Leibman, 1996) and Hardy field (Frantzikinakis-Wierdl, 2009, Frantzikinakis, 2015) extensions of the latter. In general, if the multiple average under consideration has the “expected limit”, then one obtains, via Furstenberg’s Correspondence Principle, combinatorial patterns in “large” subsets of integers.

In this talk, I will briefly present the recent topic of variable polynomials (first used in the setting of interest by Frantzikinakis, 2015, and more recently by Tsinas, 2021) and their use in the “joint ergodicity” problem (i.e., averages having the expected limit). The main part of this talk relies on a joint work with Sebastián Donoso and Wenbo Sun.