Multiple ergodic averages — Problem 17

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. Fix $\ell\ge1$ and $p_1,\ldots,p_\ell\in\mathbb Z[t]$ with $p_j(0)=0$. Rational independence means that no nonzero rational linear combination of the $p_j$ is the zero polynomial; since the constant terms vanish, this is equivalent to saying every nontrivial such combination is nonconstant.

Statement recorded in the supplied source.

For every commuting system, every $A\in\mathcal X$ and every $\varepsilon>0$, is there $n\in\mathbb N$ such that \[\mu\left(A\cap T_1^{p_1(n)}A\cap\cdots\cap T_\ell^{p_\ell(n)}A\right)\ge\mu(A)^{\ell+1}-\varepsilon?\] No ergodicity or distinct-degree hypothesis is imposed. [S2]

Short English statement

Do rationally independent zero-constant polynomials force the optimal-power recurrence lower bound in every commuting system?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800017 (AMR-047-0017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Sections 1.2 and5.2.2--5.2.3, full Problem 17, equation (18), p.27. https://arxiv.org/pdf/1103.3808v3.