Resolving the joint ergodicity problem for Hardy sequences

TL;DR

This paper advances the understanding of joint ergodicity for Hardy sequences of polynomial growth, proving the sufficiency of certain conditions and exploring the limitations of their necessity, with new theoretical tools developed.

Contribution

It establishes the sufficiency of difference and product ergodicity conditions for Hardy sequences without independence assumptions, and introduces new structural and analytical methods.

Findings

Sufficiency of difference and product conditions for joint ergodicity.

Counterexamples showing the failure of the converse for some Hardy sequences.

Development of new structural theory and analytical techniques.

Abstract

The joint ergodicity classification problem aims to characterize those sequences which are jointly ergodic along an arbitrary dynamical system if and only if they satisfy two natural, simpler-to-verify conditions on this system. These two conditions, dubbed the difference and product ergodicity conditions, naturally arise from Berend and Bergelson's pioneering work on joint ergodicity. Elaborating on our earlier work, we investigate this problem for Hardy sequences of polynomial growth, this time without making any independence assumptions on the sequences. Our main result establishes the "difficult" direction of the problem: if a Hardy family satisfies the difference and product ergodicity conditions on a given system, then it is jointly ergodic for this system. We also find that, surprisingly, the converse fails for certain pathological families of Hardy sequences, even though it holds for all "reasonable" Hardy families. We conclude by suggesting potential fixes to the statement of this problem. New ideas of independent interest developed in this paper include the structure theory of a family of factors generalizing Host-Kra and box factors; a strengthening of Tao-Ziegler's concatenation results; and the most robust extension of a seminorm smoothing argument.