Multiple ergodic averages — Problem 34

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.

Let $(X,\mathcal X,\mu)$ be a probability space. For each positive integer $q$, let $T_q$ be an invertible measure-preserving transformation, with $T_1=\operatorname{id}$ and $T_mT_n=T_{mn}$ for all $m,n\ge1$. This is a system with multiplicative structure. The harmless convention $T_0=\operatorname{id}$ is also used in the source, although no zero index occurs below. For a measurable set $A$, $T_qA$ denotes its image.

Statement recorded in the supplied source.

For every such system and every $A\in\mathcal X$ with $\mu(A)>0$, are there positive integers $m>n$ such that \[\mu\left(T_{2mn}A\cap T_{(m-n)(m+n)}A\right)>0?\] [S2]

Short English statement

Does every positive-measure set recur under two multiplicative transformations whose indices form the legs of a Pythagorean triple?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800034 (AMR-047-0034). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Section 8, multiplicative-system definition and full Problem 34, p.36. https://arxiv.org/pdf/1103.3808v3.