Ergodic averages for sparse corners

TL;DR

This paper develops a framework to analyze the limiting behavior of multiple ergodic averages with commuting transformations along sparse sequences, resolving several longstanding problems in ergodic theory and combinatorics.

Contribution

It introduces new conditions for polynomial Szemerédi theorem validity, extends results on averages along sparse sequences, and addresses open problems on correlation sequences using advanced ergodic techniques.

Findings

Necessary and sufficient conditions for polynomial Szemerédi theorem with sparse sequences.

Invariance of Furstenberg averages along sparse sequences such as [n^c].

Extension of results on common differences in polynomial and Hardy corners.

Abstract

We develop a framework for the study of the limiting behavior of multiple ergodic averages with commuting transformations when all iterates are given by the same sparse sequence; this enables us to partially resolve several longstanding problems. First, we address a special case of the joint intersectivity question of Bergelson, Leibman, and Lesigne by giving necessary and sufficient conditions under which the multidimensional polynomial Szemer\'edi theorem holds for length-three patterns. Second, we show that for two commuting transformations, the Furstenberg averages remain unchanged when the iterates are taken along sparse sequences such as $[n^c]$ for a positive noninteger $c$, advancing a conjecture of the first author. Third, we extend a result of Chu on popular common differences in linear corners to polynomial and Hardy corners. Lastly, we answer open problems of Le, Moreira, and Richter concerning decomposition results for double correlation sequences. Our toolbox includes recent degree lowering and seminorm smoothing techniques, the machinery of magic extensions of Host, and novel structured extensions motivated by works of Tao and Leng. Combined, these techniques reduce the analysis to settings where the Host-Kra theory of characteristic factors and equidistribution on nilmanifolds yield a family of striking identities from which our main results follow.