Multiple ergodic averages — Problem 35

Definitions and mathematical statement

A probability-preserving system with multiplicative structure consists of a probability space $(X,\mathcal B,\mu)$ and invertible measure-preserving transformations $T_r$, one for every positive integer $r$, such that \[ T_1=\mathrm{id},\qquad T_rT_s=T_{rs}\quad(r,s\ge1). \] Thus all the transformations commute. Equivalently, they form an action of the multiplicative positive integers by invertible maps; this extends to positive rational indices using inverses.

For every such system and every measurable $A$ with $\mu(A)>0$, must there exist positive integers $m,n$ such that \[ \mu\left( T_{m(m+n)}A\ \cap\ T_{(m+2n)(m+3n)}A\ \cap\ T_{(m+4n)(m+5n)}A \right)>0? \] All three expressions denote images of the same set under the corresponding indexed transformation. The missing $A$ after the middle transformation in the catalogue/source display is supplied explicitly.

The indices are products of consecutive terms of a single six-term arithmetic progression; they are not additive iterates $T^r$ of one fixed map. Both $m$ and $n$ must be positive, so setting $n=0$ cannot give a vacuous affirmative answer. No ergodicity, mixing, or freeness of the multiplicative action is required. The question asks only for positive measure of one three-way intersection, without prescribing a lower bound, density of good pairs, or a limit for their averages.

Short English statement

Must multiplicative actions give a positive three-way intersection at the three paired-product indices of a six-term arithmetic progression?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800035 (AMR-047-0035), 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 35. Original source indexed by AMR-047-0035. https://arxiv.org/abs/1103.3808.