Norm convergence for generalized Hardy sequences (Tsinas Conjecture 1)

Establish L^2(μ) convergence of the averages (1/N)∑_{n=1}^N T^{n^{b_1}^2}f_1⋯T^{n^{b_ℓ}^2}f_ℓ for any system (X,𝔛,μ,T), any 0<b_1<⋯<b_ℓ, and all f_1,...,f_ℓ∈L^∞(μ).

Background

This conjecture asks for a norm convergence theory for averages with iterates given by squares of fractional power functions, extending known results for standard Hardy sequences.

Such convergence would unify and extend several existing norm convergence results beyond polynomial iterates.

References

Problem [Norm convergence for generalized Hardy sequences {Conjecture 1] Let $0<b_1 < \cdots < b_\ell$, $(X, , \mu,T)$ be a system, and $f_1, \ldots, f_\ell\in L^{\infty(\mu)$.} Does the average \begin{align*} {n\in[N]}T^{{n^{{b_1}^2}f_1\cdots}} T^{{n^{b}\ell}^2}f_\ell \end{align*} converge in $L^2(\mu)$?

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

Norm convergence for generalized Hardy sequences (Tsinas Conjecture 1)

Background

References

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