Ergodic averages for sparse corners
Nikos Frantzikinakis, Borys Kuca
TL;DR
This work develops a cohesive framework for the limiting behavior of multi-ergodic averages with commuting transformations when all iterates follow a single sparse sequence. By integrating box seminorms, Host–Kra characteristic factors, nilmanifolds, Hardy-field technology, and magic/structured extensions, the authors establish limiting formulas for Hardy-field and polynomial sequences, derive Szemerédi-type recurrence consequences, and obtain nil–plus–null decompositions for double correlation sequences. Key contributions include a Hardy-field limiting formula (and prime-extended variants), a diagonal polynomial-corner limiting identity, and a suite of decomposition and recurrence results for length-three patterns under joint intereсtivity. The results advance understanding of sparse-iterate averages in the commuting-transform setting and provide robust tools (degree lowering, structured extensions, and finitary box norms) with potential applications to broader recurrence and combinatorial problems.
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 Szemerédi 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.