Multiple ergodic averages — Problem 4

Definitions and mathematical statement

Fix $\ell\ge1$ and an integer sequence $a(n)$, $n\ge1$. A connected $\ell$-step nilmanifold is a compact connected quotient $G/\Gamma$, where $G$ is a nilpotent Lie group of nilpotency class at most $\ell$ and $\Gamma$ is a discrete cocompact subgroup. Equip it with its invariant probability $m$. Call $b\in G$ irrational when $(b^n\Gamma)$ is equidistributed: the averages of every continuous function along that orbit converge to its integral against $m$.

Assume two conditions. For every such nilmanifold and every irrational $b$, the sequence $(b^{a(n)}\Gamma)$ is also equidistributed. Moreover, for every integer $r\ge1$, \[ \limsup_{N\to\infty}\frac1N \#\{1\le n\le N:r\mid a(n)\}>0. \] Prove that for every probability space $(X,\mathcal B,\mu)$, every commuting family of invertible measure-preserving maps $T_1,\ldots,T_\ell$, and every measurable $A$ with $\mu(A)>0$, some $n\ge1$ with $a(n)\ne0$ satisfies \[ \mu\left(A\cap T_1^{a(n)}A\cap\cdots \cap T_\ell^{a(n)}A\right)>0. \] Here $T^kA$ is the image of $A$ under the invertible iterate. The systems need not be ergodic. The nilmanifold tests concern all continuous observables and all eligible rotations; the positive divisibility densities may depend on $r$. Their role is a sufficient criterion for recurrence in arbitrary commuting systems, not only the nilsystems used as tests.

Short English statement

Do equidistribution on all relevant nilmanifolds and positive divisibility densities force multiple recurrence for commuting transformations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800004 (AMR-047-0004), 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 4. Original source indexed by AMR-047-0004. https://arxiv.org/abs/1103.3808.