Multiple ergodic averages — Problem 8

Definitions and mathematical statement

An integer sequence $a(n)>0$ is fast growing here if \[ \lim_{n\to\infty}\frac{\log a(n)}{\log n}=+\infty, \] or equivalently $a(n)/n^k\to\infty$ for every positive integer $k$. Construct an explicit such sequence with the following two universal properties for every $\ell\ge1$.

For every probability space $(X,\mathcal B,\mu)$ and commuting invertible measure-preserving transformations $T_1,\ldots,T_\ell$, all functions $f_i\in L^\infty(\mu)$ give convergent averages \[ \frac1N\sum_{n=1}^N\prod_{i=1}^{\ell} f_i\circ T_i^{a(n)} \quad\text{in }L^2(\mu). \] For the same class of systems, every measurable $A$ with $\mu(A)>0$ must satisfy \[ \mu\left(A\cap T_1^{a(n)}A\cap\cdots \cap T_\ell^{a(n)}A\right)>0 \quad\text{for some }n\ge1. \] Both requirements include the powers-of-one-map versions: take $T_i=R^{k_i}$, with arbitrary integer $k_i$ for convergence and nonzero $k_i$ for recurrence.

The sequence must be fixed independently of the system, the number of transformations, and the tested functions or set. "Explicit" requests a specified deterministic construction or formula together with proofs of these properties, rather than a sequence chosen separately for each system. No formula for the mean limit, and no pointwise convergence assertion, is imposed. Growth alone is insufficient; the recurrence and convergence conditions are essential parts of the requested example.

Short English statement

Construct an explicit sequence growing faster than every power which still supports all finite-order commuting recurrence and mean convergence.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800008 (AMR-047-0008), 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 8. Original source indexed by AMR-047-0008. https://arxiv.org/abs/1103.3808.