Multiple ergodic averages — Problem 24

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. Choose distinct $a,b\in(0,\infty)\setminus\mathbb Z$. Here $\lfloor t\rfloor$ denotes the largest integer at most $t$.

Statement recorded in the supplied source.

Does every ergodic system and every pair $f,g\in L^\infty(\mu)$ satisfy \[\lim_{N\to\infty}\frac1N\sum_{n=1}^N f(T^{\lfloor n^a\rfloor}x)g(T^{\lfloor n^b\rfloor}x)=\left(\int_X f\,d\mu\right)\left(\int_X g\,d\mu\right)\] for $\mu$-almost every $x$? [S2]

Short English statement

Do two distinct positive noninteger powers give almost-everywhere convergence to the product of means in every ergodic system?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800024 (AMR-047-0024). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Section 6.1.3, full Problem 24, p.32. https://arxiv.org/pdf/1103.3808v3.