Multiple ergodic averages — Problem 6

Definitions and mathematical statement

Let $a:\mathbb N\to\mathbb Z$ be an arbitrary sequence. Say that it is good for two-fold convergence of powers if for every probability space $(X,\mathcal B,\mu)$, every invertible measure-preserving transformation $R$, every pair of integers $r,s$, and every $f,g\in L^\infty(\mu)$, the averages \[ \frac1N\sum_{n=1}^N (f\circ R^{r a(n)})(g\circ R^{s a(n)}) \] converge in the norm of $L^2(\mu)$ as $N\to\infty$. Integer powers, including negative and zero powers, use invertibility and the identity convention $R^0=\mathrm{id}$.

Prove that this hypothesis implies two-fold convergence for commuting transformations: for every probability space, every commuting invertible measure-preserving $T,S$, and every bounded $f,g$, the averages \[ \frac1N\sum_{n=1}^N (f\circ T^{a(n)})(g\circ S^{a(n)}) \] converge in $L^2(\mu)$.

No ergodicity, mixing, growth, monotonicity, or distinctness assumption is made on the sequence or the transformations. The limits may depend on the system and functions; no particular limit formula is required. The hypothesis must hold universally over all single transformations and all integer pairs, rather than just for the pair of powers $R,R^2$. Commuting convergence immediately implies the hypothesis by choosing $T=R^r$, $S=R^s$; the question asks whether this necessary condition is also sufficient.

Short English statement

Does universal mean convergence along powers of one transformation already imply mean convergence for any commuting pair?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800006 (AMR-047-0006), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Frantzikinakis - Some open problems on multiple ergodic averages (2016), Problem 6. Original source indexed by AMR-047-0006. https://arxiv.org/abs/1103.3808.