Multiple ergodic averages — Problem 29

Definitions and mathematical statement

Fix a real number $c>1$ which is not an integer, and put $a(n)=\lfloor n^c\rfloor$ for $n\ge1$. Let $T_1,\ldots,T_\ell$ be commuting invertible measure-preserving maps on a probability space $(X,\mathcal B,\mu)$, with arbitrary $\ell\ge1$. For bounded measurable $f_1,\ldots,f_\ell$, define \[ A_N=\frac1N\sum_{n=1}^N\prod_{i=1}^\ell f_i\circ T_i^{a(n)}, \qquad B_N=\frac1N\sum_{n=1}^N\prod_{i=1}^\ell f_i\circ T_i^n. \] Prove that $A_N$ converges in $L^2(\mu)$ and that its limit equals the $L^2$ limit of $B_N$. Equivalently, in addition to existence of the ordinary mean limit, prove $\|A_N-B_N\|_2\to0$.

The associated recurrence assertion is that every measurable $A$ of positive measure admits some $n\ge1$ for which \[ \mu\left(A\cap T_1^{a(n)}A\cap\cdots \cap T_\ell^{a(n)}A\right)>0. \] All systems and all finite families are included, without ergodicity, mixing, or independence assumptions. The limit need not be a product of integrals; it is the ordinary commuting average's own limit. The exponent $c$ is fixed before the limit in $N$ is taken, with no uniformity as $c$ varies required. Integer powers $c$ are excluded from this particular assertion.

Short English statement

Do noninteger power times give the same mean limits and the same universal recurrence as ordinary times for commuting transformations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800029 (AMR-047-0029), 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 29. Original source indexed by AMR-047-0029. https://arxiv.org/abs/1103.3808.