Multiple ergodic averages — Problem 21

Definitions and mathematical statement

Fix $\ell\ge1$ and distinct polynomials $p_1,\ldots,p_\ell\in\mathbb Z[t]$ with $p_i(0)=0$. Let $T_1,\ldots,T_\ell$ be invertible measure-preserving transformations of a common probability space $(X,\mathcal B,\mu)$. Unlike many multiple-recurrence questions, no commutation relations between the transformations are imposed.

Prove that for every measurable $A\subset X$ of positive measure there are positive integers $m,n$ such that \[ \mu\left(A\cap T_1^{m+p_1(n)}A\cap\cdots \cap T_\ell^{m+p_\ell(n)}A\right)>0. \] The same $m$ and $n$ must be used for every transformation. For an integer $k$, the notation $T_i^kA$ denotes the image under the corresponding invertible iterate; values of $m+p_i(n)$ may have either sign or be zero.

Distinctness means that no two polynomials agree identically. Since all constant terms vanish, their pairwise differences are nonconstant. No further linear independence condition on the polynomial family, and no ergodicity or mixing condition on the system, is assumed. Both positive parameters may depend on $A$ and on the entire input data. The target is one positive simultaneous intersection, not a particular limiting formula or uniform lower bound. The additional common offset $m$ is part of the requested assertion and cannot be removed when formulating this noncommuting version.

Short English statement

Does a common extra time offset restore polynomial multiple recurrence for transformations that need not commute?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800021 (AMR-047-0021), 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 21. Original source indexed by AMR-047-0021. https://arxiv.org/abs/1103.3808.