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Tag Archives: Furstenberg
Szemerédi’s Theorem Part V – Compact extensions
This is the fifth in a series of six posts I am writing about Szemerédi’s theorem. In the previous post I proved that the Sz property lifts through weak mixing extension and in this post I will prove that the … Continue reading
Posted in Ergodic Theory
Tagged almost periodic, Compact extensions, Furstenberg, Szemeredi's theorem
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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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Szemerédi’s Theorem Part I – Equivalent formulations
The theorem of van der Waerden on arithmetic progressions, whose precise statement and proof can be found in a previous post of mine, states that in a finite partition of the set of positive integers, one of the pieces contains … Continue reading
Posted in Combinatorics, Ergodic Theory, Ramsey Theory
Tagged Furstenberg, Szemeredi's theorem
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Convergence and Recurrence of Z actions
Ergodic Ramsey Theory started with Furstenberg’s proof of Szemeredi’s theorem in arithmetic progressions in 1977. Through a correspondence principle, Furstenberg realized that Szemeredi’s theorem follows from a dynamical statement: for every invertible, ergodic measure preserving transformation of a probability space … Continue reading
Posted in Ergodic Theory, State of the art
Tagged Austin, Bergelson, convergence, Conze, ergodic theorem, Furstenberg, Host, Katznelson, Kra, Leibman, Lesigne, Poincare, recurrence, Tao, Walsh
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Sets of nice recurrence
— 1. Introduction — Let be a probability space and be a (measurable) map such that the set has the same measure as the set for all (measurable) sets . We call the triple a measure preserving system. All sets … Continue reading
Posted in Combinatorics, Ergodic Theory
Tagged Banach density, Correspondence principle, Furstenberg, nice recurrence, recurrence
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Furstenberg’s Correspondence Theorem
In 1977 Furstenberg gave a new proof of Szemerédi’s theorem using ergodic theory. The first step in that proof was to turn the combinatorial statement into a statement in ergodic theory. Thus Furstenberg created what is now known as Furstenberg’s … Continue reading