Doctoral theses
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Current Record: 31 of 2491
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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
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Τσίνας, Κωνσταντίνος
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Thesis advisor
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Φραντζικινάκης, Νικόλαος
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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
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Language |
English |
Subject |
Ergodic theory |
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Hardy sequences |
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Multiple recurrence |
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Ακολουθίες hardy |
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Εργοδική θεωρία |
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Πολλαπλή επαναφορά |
Issue date |
2024-07-19 |
Collection
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School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Doctoral theses
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Type of Work--Doctoral theses
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Permanent Link |
https://elocus.lib.uoc.gr//dlib/2/a/4/metadata-dlib-1720510965-301354-19643.tkl
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Views |
143 |