Multiple ergodic averages — Problem 26

Definitions and mathematical statement

A Hardy field is a field of germs at $+\infty$ of real-valued differentiable functions on half-lines, closed under differentiation. Two functions represent the same germ when they eventually agree. Seek a function $a$ belonging to some Hardy field with \[ a(t)/t^k\longrightarrow+\infty \quad\text{for every integer }k\ge1. \] Choose an eventual representative and set $b(n)=\lfloor a(n)\rfloor$ for all sufficiently large $n$; its finitely many earlier terms may be assigned positive integers.

The required sequence must have both properties below for every $\ell\ge1$ and every invertible measure-preserving $T$ on a probability space. For all integer $k_1,\ldots,k_\ell$ and bounded functions $f_i$, the averages \[ \frac1N\sum_{n=1}^N\prod_{i=1}^\ell f_i\circ T^{k_i b(n)} \] must converge in $L^2(\mu)$. For all nonzero integers $k_i$ and all $A$ with $\mu(A)>0$, some $n$ with $b(n)\ne0$ must satisfy $\mu(A\cap T^{k_1b(n)}A\cap\cdots\cap T^{k_\ell b(n)}A)>0$.

The problem is to construct one such Hardy-field function and prove these universal assertions. Membership in a Hardy field is an additional regularity requirement beyond rapid growth of an arbitrary sequence. No ergodicity or pointwise convergence is required. Convergence must hold for all integer powers, and recurrence for all nonzero powers, independently of the chosen system.

Short English statement

Find a super-polynomial Hardy-field sequence that supports multiple recurrence and mean convergence along every finite collection of powers.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800026 (AMR-047-0026), 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 26. Original source indexed by AMR-047-0026. https://arxiv.org/abs/1103.3808.