Total joint ergodicity for totally ergodic systems

TL;DR

This paper characterizes total joint ergodicity for totally ergodic systems involving polynomial iterates, providing complete results for up to two iterates and extending previous conjectures with new methods.

Contribution

It offers a full characterization of total joint ergodicity for up to two iterates and extends results on Hardy field functions, disproving a prior conjecture.

Findings

Complete characterization for at most two iterates

Disproved a conjecture of the first author

Extended results on Hardy field functions of polynomial growth

Abstract

Examining multiple ergodic averages whose iterates are integer parts of real valued polynomials for totally ergodic systems, we provide various characterizations of total joint ergodicity, meaning that an average converges to the "expected" limit along every arithmetic progression. In particular, we obtain a complete characterization when the number of iterates is at most two, and disprove a conjecture of the first author. We also improve a result of Frantzikinakis on joint ergodicity of Hardy field functions of at most polynomial growth for totally ergodic systems, which extends a conjecture of Bergelson-Moreira-Richter. Our method is to first use the methodology of Frantzikinakis, which allows one to reduce the systems to rotations on abelian groups without using deep tools from ergodic theory, then develop formulas for integrals of exponential functions over subtori, and finally, compute exponential sums for integer parts of real polynomials.