# 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

## 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 | | 4 Comments

## 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 | | 6 Comments

## 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 | | Leave a comment

## 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 | | 1 Comment

## 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

Posted in Combinatorics, Ergodic Theory, Tool | Tagged , | 13 Comments