Translator Disclaimer
May 2022 Pointwise ergodic theorems for non-conventional bilinear polynomial averages
Ben Krause, Mariusz Mirek, Terence Tao
Author Affiliations +
Ann. of Math. (2) 195(3): 997-1109 (May 2022). DOI: 10.4007/annals.2022.195.3.4

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.

Citation

Download Citation

Ben Krause. Mariusz Mirek. Terence Tao. "Pointwise ergodic theorems for non-conventional bilinear polynomial averages." Ann. of Math. (2) 195 (3) 997 - 1109, May 2022. https://doi.org/10.4007/annals.2022.195.3.4

Information

Published: May 2022
First available in Project Euclid: 29 April 2022

Digital Object Identifier: 10.4007/annals.2022.195.3.4

Subjects:
Primary: 11L03 , 11L07 , 11L15 , 11P55 , 37A30 , 37A46 , 42A45 , 42A50 , 42A85‎ , 43A25

Keywords: adelic integres , circle method , Fourier methods , inverse theorems , non-conventional ergodic averages , pointwise convergence , variational estimates

Rights: Copyright © 2022 Department of Mathematics, Princeton University

JOURNAL ARTICLE
113 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.195 • No. 3 • May 2022
Back to Top