Multiple ergodic averages — Problem 14

Definitions and mathematical statement

A generalized polynomial is a real-valued function on $\mathbb Z$ obtained from the identity function $n\mapsto n$ and real constants by finitely many additions, multiplications, and applications of the floor function. For example, $n\mapsto\lfloor\alpha n\lfloor\beta n\rfloor +\gamma n^2\rfloor$ is such a function. The number and nesting of operations are finite but otherwise unrestricted. Call it integer valued if every input integer has an integer output.

Fix $\ell\ge1$ and integer-valued generalized polynomials $p_1,\ldots,p_\ell$. Let $T_1,\ldots,T_\ell$ be commuting invertible measure-preserving transformations of a common probability space $(X,\mathcal B,\mu)$. Prove that for all functions $f_i\in L^\infty(\mu)$ the sequence \[ \frac1N\sum_{n=1}^N \prod_{i=1}^\ell f_i\circ T_i^{p_i(n)} \] converges in $L^2(\mu)$ as $N\to\infty$.

Integer outputs may be zero or negative, so negative iterates are interpreted using invertibility. The generalized polynomials need not be distinct, have zero constant term, or satisfy any independence condition. No ergodicity, mixing, or pointwise convergence is assumed or requested. The target is existence of a norm limit for every finite family, without prescribing its value. Nested floor operations are essential to the scope: restricting to ordinary polynomials, or to a single floor applied to a real polynomial, would omit the general problem.

Short English statement

Do multiple averages along arbitrary integer-valued generalized polynomials always converge in mean for commuting transformations?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800014 (AMR-047-0014), 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 14. Original source indexed by AMR-047-0014. https://arxiv.org/abs/1103.3808.