Multiple recurrence for generalized Hardy sequences (Tsinas Conjecture 3)

Prove that for any 0<b_1<⋯<b_ℓ, any system (X,𝔛,μ,T), and any E∈𝔛 with μ(E)>0, there exists n∈ℤ such that μ(E∩T^{-⌊n^{b_1}⌋^2}E∩⋯∩T^{-⌊n^{b_ℓ}⌋^2}E)>0.

Background

This conjecture seeks a Szemerédi-type multiple recurrence statement for patterns given by squares of fractional powers, extending classical polynomial recurrence.

It would provide new combinatorial consequences via the correspondence principle.

References

Problem [Multiple recurrence for generalized Hardy sequences {Conjecture 3] Let $0<b_1 < \cdots < b_\ell$, $(X, , \mu, T)$ be a system, and $E\in$ with $\mu(E)\>0$. Does there exist $n\in$ for which \begin{align*} \mu(E\cap T{-{n{b_1}2}E\cap\cdots \cap T{-{n{b_\ell}2}E)>0? \end{align*}

— Joint ergodicity - 40 years on  (2603.18974 - Kuca, 19 Mar 2026) in Section 3.5 (Generalized Hardy sequences)