Multiple ergodic averages — Problem 9

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A system is a probability space $(X,\mathcal X,\mu)$ with invertible measure-preserving transformations; multiple transformations are assumed to commute. Write $T^j f=f\circ T^j$, and $T^j A$ for the image of a measurable set under $T^j$. A system with one transformation is ergodic if every invariant measurable set has measure zero or one. The space $L^\infty(\mu)$ consists of essentially bounded complex measurable functions. Let $\mathcal P=\{p_1,\ldots,p_\ell\}\subset\mathbb Z[t]$, where each $p_j$ and each difference $p_i-p_j$ ($i\ne j$) is nonconstant. Define $A_N(f_1,\ldots,f_\ell)=N^{-1}\sum_{n=1}^N\prod_jT^{p_j(n)}f_j$. For a $T$-invariant sub-$\sigma$-algebra $\mathcal Z$, being characteristic means that replacing every $f_j$ by $\mathbb E(f_j\mid\mathcal Z)$ changes $A_N$ by a quantity tending to zero in $L^2$. Define the Host--Kra seminorms by $\|f\|_{U^1(T)}=\|\mathbb E(f\mid\mathcal I_T)\|_2$, where $\mathcal I_T$ is the invariant $\sigma$-algebra, and \[\|f\|_{U^{s+1}(T)}^{2^{s+1}}=\lim_{H\to\infty}\frac1H\sum_{h=1}^H\|\bar f\,T^h f\|_{U^s(T)}^{2^s}.\] The factor $\mathcal Z_{d,T}$, $d\ge0$, is characterized by $\mathbb E(f\mid\mathcal Z_{d,T})=0$ if and only if $\|f\|_{U^{d+1}(T)}=0$. Let $d_{\min}(\mathcal P)$ be the least nonnegative $d$ such that $\mathcal Z_{d,T}$ is characteristic for these averages in every system.

Statement recorded in the supplied source.

For every essentially distinct family with $\ell=|\mathcal P|\ge2$, is $d_{\min}(\mathcal P)\le\ell-1$? [S2]

Short English statement

Can the necessary Host--Kra factor order always be bounded by one less than the number of polynomials, independently of their degrees?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800009 (AMR-047-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Sections 2.2--2.5,5.1 and5.1.1; equation (16), full Problem 9, pp.4--6,22--23. https://arxiv.org/pdf/1103.3808v3.