Multiple ergodic averages — Problem 18

Definitions and mathematical statement

Let $p_1,\ldots,p_\ell$ be integer polynomials, with $\ell\ge1$. The family is jointly intersective if \[ \forall r\ge1\quad\exists n\ge1\quad p_i(n)\equiv0\pmod r\quad(1\le i\le\ell). \] The same $n$ must work for every polynomial at the chosen modulus; requiring separate roots for each polynomial would be weaker.

Let $(X,\mathcal B,\mu)$ be any probability space and $T_1,\ldots,T_\ell$ any commuting invertible measure-preserving transformations. Is joint intersectivity sufficient to ensure that every $A\in\mathcal B$ with $\mu(A)>0$ has \[ \mu\left(A\cap T_1^{p_1(n)}A\cap\cdots \cap T_\ell^{p_\ell(n)}A\right)>0 \quad\text{for some }n\ge1? \] Here $T_i^kA$ is the image under the invertible iterate, also when $k$ is negative. The positive integer $n$ may depend on the entire system and on $A$.

No zero-constant-term, distinctness, rational independence, or ergodicity hypothesis is imposed. In particular, intersective polynomials need not have a common integer root. The requested conclusion is positive measure of one simultaneous intersection, without a prescribed quantitative lower bound. The author's progress page records a result for two polynomials; the general arbitrary-family target is retained here. Testing periodic systems explains the modular obstruction, but the question asks for sufficiency in all probability-preserving systems.

Short English statement

Do common roots modulo every integer guarantee polynomial multiple recurrence for arbitrary commuting transformations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800018 (AMR-047-0018), 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 18. Original source indexed by AMR-047-0018. https://arxiv.org/abs/1103.3808.
• S3 Frantzikinakis, November 2025 progress update for Problem 18. https://users.math.uoc.gr/~frantzikinakis/OpenProblemsNew.html.