Multiple ergodic averages — Problem 22

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A system is a probability space $(X,\mathcal X,\mu)$ with invertible measure-preserving transformations; multiple transformations are assumed to commute. Write $T^j f=f\circ T^j$, and $T^j A$ for the image of a measurable set under $T^j$. A system with one transformation is ergodic if every invariant measurable set has measure zero or one. The space $L^\infty(\mu)$ consists of essentially bounded complex measurable functions. A Hardy field is a field of germs at $+\infty$ of real differentiable functions, closed under differentiation; two functions have the same germ if they agree eventually. Let $\mathcal F=\{a_1,\ldots,a_\ell\}$ lie in one such field and have polynomial growth, meaning $a_j(t)/t^k\to0$ for some integer $k\ge1$. Write $\operatorname{span}^*(\mathcal F)$ for the nonzero real linear combinations of its elements. Integer parts are floors; changing finitely many initial values has no effect on convergence. Being good for $\ell$-convergence of a single transformation means that for every system and all bounded $f_j$, $N^{-1}\sum_{n=1}^N\prod_jT^{\lfloor a_j(n)\rfloor}f_j$ converges in $L^2(\mu)$.

Statement recorded in the supplied source.

Is this convergence property equivalent to requiring that each $a\in\operatorname{span}^*(\mathcal F)$ satisfy at least one of:

1. $|a(t)-cp(t)|/\log t\to\infty$ for every $c\in\mathbb R$ and $p\in\mathbb Z[t]$;
2. $a(t)-cp(t)\to d$ for some $c,d\in\mathbb R$ and some $p\in\mathbb Z[t]$;
3. $|a(t)-t/m|\le C\log t$ for all sufficiently large $t$, for some $m\in\mathbb Z\setminus\{0\}$ and $C>0$.

[S2]

Short English statement

Do these three alternatives for every nonzero linear combination exactly characterize universal mean convergence of Hardy-field multiple averages?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800022 (AMR-047-0022). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Section 6 introduction and6.1--6.1.1; equation (20), preceding single-function theorem and full Problem 22, pp.30--31. https://arxiv.org/pdf/1103.3808v3.