On the norm convergence of non-conventional ergodic averages

Abstract

We offer a proof of the following nonconventional ergodic theorem:The leading case of this result, with r = 1 and the standard sequence of averaging sets, was first proved by Tao in [16], following earlier analyses of various more special cases and related results by Conze and Lesigne [4,5,6], Furstenberg and Weiss [9], Zhang [18], Host and Kra [12,13], Frantzikinakis and Kra [7] and Ziegler [19]. While Tao's proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

Ergodic averages of commuting transformations with distinct degree polynomial iterates

“…Unlike previous arguments in [11,13,31,32,2,22], where one finds ways to sidestep the problem of giving precise algebraic descriptions of the factor systems that control the limiting behavior of special cases of the averages (2), a distinctive feature of the proof of Theorem 1.1 is that we give such descriptions.…”

“…1 Furthermore, we did not find it advantageous to work within a suitable extension of our system in order to simplify our study (like the "pleasant" or "magic" extensions that were introduced in [2] and in [22] respectively). In this respect, our analysis is more closely related to the one used to study convergence results when all the transformations T 1 , .…”

“…3 It is in cases like this that working with some kind of "pleasant" extension (using terminology from [2]) or "magic" extension (using terminology form [22]) of the system may offer an essential advantage (this is indeed the case when all the polynomials are linear).…”

Анализ Фурье В Комбинаторной Теории Чисел

“…Мы найдем подпространство H и вектор x такие, что n, то мы найдем монохроматическое решение уравнения (37). Вспоминая теперь, что k = R(c), получаем оценку (35). Для доказательства неравенства (36) до-статочно заметить, что в случае l log n формула (40) переписывается в виде…”

“…Следует отметить, что, используя свой метод, Фюрстенберг и его ученики получили множество глубоких обобщений теоремы Семереди (см., например, [24]- [35], [38], [39], [86]- [88]), которые комбинаторными методами доказать пока не удалось. Подробнее о теореме Семереди можно узнать в обзоре [4].…”