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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	School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Doctoral theses
	Type of Work--Doctoral theses
Permanent Link	https://elocus.lib.uoc.gr//dlib/2/a/4/metadata-dlib-1720510965-301354-19643.tkl
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