10.3 Furstenberg's ergodic theoretical proof

Furstenberg's proof of Szemerédi's theorem is a game-changer in math. It uses ergodic theory to solve a tricky combinatorial problem, showing that sets of integers with positive upper density have long arithmetic progressions.

This approach connects ergodic theory and combinatorics in a new way. It introduces multiple recurrence and the correspondence principle, which have become key tools in both fields. Furstenberg's work sparked a whole new area of study: ergodic Ramsey theory.

Furstenberg's Proof Steps

Translating the Combinatorial Problem

Top images from around the web for Translating the Combinatorial Problem

• 

  Ergodic stationary process | The Blue Dot View original

  Is this image relevant?

• 

  Ergodic theory - Wikipedia View original

  Is this image relevant?

• 

  Ergodic stationary process | The Blue Dot View original

  Is this image relevant?

• 

  Ergodic stationary process | The Blue Dot View original

  Is this image relevant?

• 

  Ergodic theory - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Translating the Combinatorial Problem

• 

  Ergodic stationary process | The Blue Dot View original

  Is this image relevant?

• 

  Ergodic theory - Wikipedia View original

  Is this image relevant?

• 

  Ergodic stationary process | The Blue Dot View original

  Is this image relevant?

• 

  Ergodic stationary process | The Blue Dot View original

  Is this image relevant?

• 

  Ergodic theory - Wikipedia View original

  Is this image relevant?

1 of 3

• Szemerédi's theorem states that any set of integers with positive upper density contains arbitrarily long arithmetic progressions
• Furstenberg's proof translates the combinatorial problem into the language of ergodic theory by considering the shift operator on the space of configurations
• The proof establishes a correspondence between sets of integers with positive upper density and shift-invariant measures on the space of configurations

Introducing Multiple Recurrence

• Furstenberg introduces the notion of multiple recurrence, which states that for any measure-preserving system and any set of positive measure, almost every point returns to the set infinitely often under iterations of the transformation
• The proof shows that multiple recurrence implies the existence of arithmetic progressions in sets of positive upper density
• The key step in the proof is the use of the ergodic decomposition and the correspondence principle to reduce the problem to the case of ergodic measures
• For ergodic measures, multiple recurrence is proved using the Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem

Ergodic Measures and Recurrence

Properties of Ergodic Measures

• Ergodic measures are shift-invariant measures that cannot be decomposed into non-trivial invariant components
• They play a crucial role in the proof because they exhibit strong recurrence properties
• The Furstenberg-Zimmer structure theorem is used to decompose the space into ergodic components, reducing the problem to the case of ergodic measures

Multiple Recurrence and Arithmetic Progressions

• Multiple recurrence is a generalization of Poincaré recurrence, stating that for any measure-preserving system and any set of positive measure, almost every point returns to the set infinitely often under iterations of the transformation
• Furstenberg's proof establishes a connection between multiple recurrence and the existence of arithmetic progressions in sets of positive upper density
• The Furstenberg recurrence theorem, which is a consequence of the von Neumann mean ergodic theorem, is applied to prove multiple recurrence for ergodic measures
• The correspondence principle then allows the transfer of the multiple recurrence result back to the combinatorial setting, proving Szemerédi's theorem (existence of arithmetic progressions in sets of positive upper density)

Significance of Furstenberg's Proof

Connecting Ergodic Theory and Combinatorics

• Furstenberg's proof introduced a new approach to combinatorial problems by applying methods from ergodic theory, opening up a fruitful connection between the two fields
• The proof demonstrated the power of ergodic theory in tackling problems in additive combinatorics and Ramsey theory, leading to the development of ergodic Ramsey theory
• Ergodic Ramsey theory studies the recurrence properties of measure-preserving systems and their applications to combinatorial problems, extending the ideas introduced in Furstenberg's proof

Impact on Further Research

• Furstenberg's proof inspired further research on the connections between ergodic theory and combinatorics, leading to the development of new tools and techniques in both fields
• The concepts and methods introduced in Furstenberg's proof, such as multiple recurrence and the correspondence principle, have found applications in various areas of mathematics (number theory, harmonic analysis)
• The success of Furstenberg's approach has led to the discovery of new combinatorial results and the strengthening of existing ones, showcasing the fruitful interplay between ergodic theory and combinatorics

Furstenberg's Proof vs Others

Szemerédi's Original Proof (1975)

• Szemerédi's original proof was combinatorial and used the regularity lemma, which decomposes a graph into a structured part and a pseudo-random part
• In contrast, Furstenberg's proof (1977) uses ergodic theory and introduces the concept of multiple recurrence

Gowers' Proof (2001)

• Gowers' proof uses a combination of combinatorial and Fourier-analytic techniques, relying on the notion of uniformity norms and the dichotomy between structure and randomness
• While different in approach, both Furstenberg's and Gowers' proofs exploit the idea of decomposing the problem into structured and random components

Hypergraph Removal Lemma Proof (2006)

• The hypergraph removal lemma proof, developed by Nagle, Rödl, Schacht, and Skokan, uses a generalization of the regularity lemma to hypergraphs
• This proof is combinatorial in nature, unlike Furstenberg's ergodic-theoretic approach

Polymath Proof (2012)

• The polymath proof, a collaborative effort by several mathematicians, combines ideas from ergodic theory and Fourier analysis
• It shares some similarities with Furstenberg's proof in its use of ergodic-theoretic concepts but also incorporates Fourier-analytic techniques