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Doctoral theses

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Identifier

000465967

Title

Multiple ergodic averages along sequences of polynomial growth

Alternative Title

Πολλαπλοί εργοδικοί μέσοι για ακολουθίες πολυωνομικού ρυθμού αύξησης

Author

Τσίνας, Κωνσταντίνος

Thesis advisor

Φραντζικινάκης, Νικόλαος

Abstract

Ergodic theory has been an active area of research in recent decades. Furstenberg’s original work in the proof of Szemer´edi’s theorem was the spark for the development of a whole new research field, since dynamical methods were then understood to be a potent tool for solving combinatorial problems. These methods naturally led researchers to ask many follow-up questions and nowadays we have generalizations of Furstenberg’s results concerning patterns beyond arithmetic progressions as well as deep theorems describing the structure of measure-preserving systems. In this thesis, we investigate the problem of convergence of multiple ergodic averages along sequences that arise from functions that have polynomial growth and some extra regularity properties, such as monotonicity and smoothness. Typical examples include the polynomials or the fractional powers nc, where c > 0 is not an integer. We show that under some simple assumptions on the growth rates of the functions we have convergence of multiple ergodic averages along these sequences in all measure-preserving systems. As a consequence of these results, we derive several combinatorial applications showing that all subsets of Z with positive density contain patterns of a specific form. In the case of nilmanifolds, we prove pointwise convergence results for these averages and then use well-known structure theorems to deduce convergence results for general measure-preserving systems. Furthermore, we ask the same questions for multiple ergodic averages evaluated along the prime numbers and we show that under the same assumptions, the corresponding averages converge and the limit is the same as the limit of the typical averages along the naturals. The results of this thesis are contained in the following articles (listed in chronological order): 1) K. Tsinas. Joint ergodicity of Hardy field sequences. Transactions of the American Mathematical Society, 376:3191–3263, 2023. 2) K. Tsinas. Pointwise convergence in nilmanifolds along smooth functions of polynomial growth. Ergodic Theory and Dynamical Systems. Published online p:1-46. doi:10.1017/etds.2023.6, 2023 3) A. Koutsogiannis and K. Tsinas. Ergodic averages for sparse sequences along primes. Preprint 2023, arXiv.2309.0493

Language

English

Subject

Ergodic theory

Hardy sequences

Multiple recurrence

Ακολουθίες hardy

Εργοδική θεωρία

Πολλαπλή επαναφορά

Issue date

2024-07-19

Collection