Weighted classification for Hardy sequences (Conjecture 8.8)

Develop a weight W with 1≺W(t)≪t such that, for any Hardy sequences a_1,...,a_ℓ of polynomial growth, the joint ergodicity classification (equivalence to product and difference ergodicity conditions) holds for W-weighted multiple ergodic averages on all measure-preserving systems.

Background

The standard classification via product/difference ergodicity fails for certain Hardy sequences in the unweighted setting.

A conjectural remedy is to adopt appropriately chosen weights W to restore an equivalence for weighted averages.

References

Fix $a_1, \ldots, a_\ell\in$. Can we always find $1 \prec W(t) \ll t$ such that Problem \ref{Pr: joint ergodicity problem} admits affirmative answer for $a_1, \ldots, a_\ell$ if we define joint ergodicity as well as product and difference ergodicity conditions using $W$-averaging schemes?

— Joint ergodicity - 40 years on  (2603.18974 - Kuca, 19 Mar 2026) in Section 4.3 (Fixes to the classification problem)