Pointwise ergodic theorems for non-conventional bilinear polynomial averages

Abstract

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages
\[
A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x)
$$\] as $N \to \infty $, where $T\colon X\to X$ is a measure-preserving transformation of a $\sigma $-finite measure space $(X,\mu )$, $P(\mathrm {n}) \in \mathbb Z[\mathrm {n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X), g \in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $\frac {1}{p_1} + \frac {1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to “break duality” by handling some ranges of exponents $p_1,p_2$ with $\frac {1}{p_1}+\frac {1}{p_2} > 1$, at the cost of increasing $r$ slightly.

This gives an affirmative answer to Problem 11 from Frantzikinakis’ open problems survey for the Furstenberg–Weiss averages (with $P(\mathrm {n})=\mathrm {n}^2$), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. This also gives a contribution to the Furstenberg–Bergelson–Leibman conjecture. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $\mathbb A_{\mathbb Z}$ also plays a role.