# A new proof of Szemerédi's theorem

@article{Gowers2001ANP,
title={A new proof of Szemer{\'e}di's theorem},
author={William T. Gowers},
journal={Geometric \& Functional Analysis GAFA},
year={2001},
volume={11},
pages={465-588}
}
• W. T. Gowers
• Published 15 August 2001
• Mathematics
• Geometric & Functional Analysis GAFA
In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these classes contains arbitrarily long arithmetic progressions. This is one of the fundamental results of Ramsey theory, and it has been strengthened in many different directions. A more precise statement of the theorem is as follows.
798 Citations
Szemerédi's theorem and problems on arithmetic progressions
Szemeredi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem
Yet Another Proof Of Szemerédi's Theorem
• Mathematics
• 2010
To Endre Szemeredi on the occasion of his 70th birthday Using the density-increment strategy of Roth and Gowers, we derive Szemeredi’s theorem on arithmetic progressions from the inverse conjectures
Arithmetic Progressions and Ergodic Theory
• Mathematics
• 2015
In Chapter 19 we saw van der Waerden’s theorem (1927) as an application of topological dynamics: If we color the natural numbers with finitely many colors, then we find arbitrarily long monochromatic
Squares in Arithmetic Progressions and Infinitely Many Primes
Abstract We give a new proof that there are infinitely many primes, relying on van derWaerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an
New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
• Mathematics
• 2012
Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was
Szemerédi's Theorem in the Primes
• Mathematics
Proceedings of the Edinburgh Mathematical Society
• 2018
Abstract Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any
On a Generalization of Szemerédi's Theorem
Let N be a natural number and A ⊂ [1, …, N]2 be a set of cardinality at least N2/(loglog⁡N)c is an absolute constant. We prove that A contains a triple {(k, m), (k+d, m), (k, m+d)}, where d > 0. This
COMBINATORIAL AND ERGODIC APPROACHES TO SZEMERÉDI 3 where
A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality , we know of four types of
The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view
A long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput,
Erdős Semi-Groups, Arithmetic Progressions, and Szemerédi's Theorem
• Han Yu
• Mathematics
Real Analysis Exchange
• 2019
In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.

## References

SHOWING 1-10 OF 33 REFERENCES
The ergodic theoretical proof of Szemerédi's theorem
• Mathematics
• 1982
Partial results were obtained previously by K. F. Roth (1952) who established the existence of arithmetic progressions of length three in subsets of Z of positive upper density, and by E. Szemeredi
Lower bounds of tower type for Szemerédi's uniformity lemma
This paper shows that the bound is necessarily of tower type, obtaining a lower bound given by a tower of 2s of height proportional to $\log{(1/ \epsilon)}$).
On sets of integers containing k elements in arithmetic progression
In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic
Integer Sets Containing No Arithmetic Progressions
lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of
Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
• Mathematics
• 1996
An extension of the classical van der Waerden and Szemeredi the- orems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following
Additive Number Theory: Inverse Problems and the Geometry of Sumsets
Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA
The Hardy-Littlewood Method
1. Introduction and historical background 2. The simplest upper bound for G(k) 3. Goldbach's problems 4. The major arcs in Waring's problem 5. Vinogradov's methods 6. Davenport's methods 7.
Primitive recursive bounds for van der Waerden numbers
a = 1, a+ = 2U(n,c '), i > O. Then U(n + ?, c) = ac . In order to get some idea of the rate of growth of U(n, c) (and of the functions which will be introduced later in this paper), we shall define
Some sequences of integers