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Tag Archives: weak mixing
Erdős Sumset conjecture
Hindman’s finite sums theorem is one of the most famous and useful theorems in Ramsey theory. It states that for any finite partition of the natural numbers, one of the cells of this partition contains an IPset, i.e., there exists … Continue reading
Posted in Combinatorics, Number Theory, State of the art
Tagged Austin, Banach density, Bohr sets, erdos, Jin, sumset, Ultrafilters, weak mixing
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Szemerédi Theorem Part VI – Dichotomy between weak mixing and compact extension
This is the sixth and final post in a series about Szemerédi’s theorem. In this post I complete the proof of the Multiple Recurrence Theorem, which I showed in a previous post of this series to be equivalent to Szemerédi’s … Continue reading
Posted in Ergodic Theory, Ramsey Theory
Tagged Compact extensions, joinings, Szemeredi's theorem, weak mixing
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Szemerédi’s Theorem Part IV – Weak mixing extensions
This is the fourth in a series of six posts I am writing about Szemerédi’s theorem. In the first three posts, besides setting up the notation and definitions necessary, I reduced Szemerédi’s theorem to three facts. Those three facts are … Continue reading
Szemerédi’s Theorem Part III – Precise definitions
This is the third in a series of six posts on Szemerédi’s theorem. In the previous post I outlined the ideas of the ergodic theoretical proof by Furstenberg. In this post I will set up the machinery and give the … Continue reading
Posted in Combinatorics, Ergodic Theory, Ramsey Theory
Tagged almost periodic, skew product, Szemeredi's theorem, weak mixing
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Szemerédi’s Theorem Part II – Overview of the proof
This is the second in a series of posts about Szemerédi’s theorem. In the first post I presented the first step in the proof of Szemerédi’ theorem, namely applying the correspondence principle of Furstenberg to transform the problem into one … Continue reading
Posted in Combinatorics, Ergodic Theory
Tagged almost periodic, cesaro limit, Extension, factor, Furstenberg, Syndetic sets, Szemerédi, Szemeredi's theorem, weak mixing
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Weak Mixing
— 1. Introduction — When studying measure preserving systems (defined below) there are many important classes that are worth studying separately. One way to distinguish between different classes is the level of “mixing” or “randomness” of the system. In this … Continue reading
Posted in Analysis, Ergodic Theory
Tagged idempotents, minimal, Ultrafilters, van der Corput, weak mixing
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Koopmanvon Neumann Decomposition
In my previous post I presented an ergodic theoretical proof of Roth’s Theorem, assuming the Koopmanvon Neumann Decomposition (and some other minor facts). In this post I present a proof of this Decomposition and moreover prove that the compact vectors … Continue reading