Multiple ergodic averages — Problem 10

Definitions and mathematical statement

Fix $\ell\ge1$. For each $N\ge1$ let $p_{1,N},\ldots,p_{\ell,N}\in\mathbb R[t]$, with degrees bounded by one finite constant independent of $i,N$. Call this array good if for every nonzero vector $c=(c_1,\ldots,c_\ell)\in\mathbb R^\ell$, \[ \lim_{N\to\infty}\frac1N\sum_{n=1}^N \exp\!\left(2\pi i\sum_{j=1}^\ell c_jp_{j,N}(n)\right)=0. \] This is the source's requirement that every nontrivial real linear combination be a good polynomial sequence; the coefficients $c_j$ are fixed as $N$ varies.

Let $T$ be any invertible ergodic measure-preserving transformation of a probability space $(X,\mathcal B,\mu)$. Prove that for every $f_1,\ldots,f_\ell\in L^\infty(\mu)$, \[ \frac1N\sum_{n=1}^N\prod_{j=1}^\ell f_j\circ T^{\lfloor p_{j,N}(n)\rfloor} \longrightarrow\prod_{j=1}^\ell\int_X f_j\,d\mu \quad\text{in }L^2(\mu). \] The floor denotes the greatest integer at most its real argument, including for negative values. Ergodicity means that invariant measurable sets have measure zero or one; total ergodicity is not assumed. Polynomial coefficients may vary with $N$, without separate size bounds. The same upper index $N$ specifies both the polynomial row and the averaging interval. The conclusion asks for the displayed product of integrals, not merely existence of some limit.

Short English statement

Does joint exponential-sum cancellation for variable polynomials force the corresponding ergodic averages to converge to the product of means?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800010 (AMR-047-0010), 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 10. Original source indexed by AMR-047-0010. https://arxiv.org/abs/1103.3808.