A Quantitative Ergodic Theory Proof of Szemerédi's Theorem

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemer\'edi's original combinatorial proof using the Szemer\'edi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers' more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is ``elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

[1]  Terence Tao,et al.  The Gaussian primes contain arbitrarily shaped constellations , 2005 .

[2]  H. Furstenberg,et al.  A density version of the Hales-Jewett theorem , 1991 .

[3]  Saharon Shelah,et al.  Primitive recursive bounds for van der Waerden numbers , 1988 .

[4]  H. Furstenberg,et al.  An ergodic Szemerédi theorem for commuting transformations , 1978 .

[5]  T. Tao,et al.  The primes contain arbitrarily long arithmetic progressions , 2004, math/0404188.

[6]  Vitaly Bergelson,et al.  Polynomial extensions of van der Waerden’s and Szemerédi’s theorems , 1996 .

[7]  Vojtech Rödl,et al.  Regularity Lemma for k‐uniform hypergraphs , 2004, Random Struct. Algorithms.

[8]  Ramsey Theory,et al.  Ramsey Theory , 2020, Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic.

[9]  W. T. Gowers,et al.  Lower bounds of tower type for Szemerédi's uniformity lemma , 1997 .

[10]  Peter J. Cameron,et al.  Some sequences of integers , 1989, Discret. Math..

[11]  W. T. Gowers,et al.  Hypergraph regularity and the multidimensional Szemerédi theorem , 2007, 0710.3032.

[12]  Matthew Foreman,et al.  The complexity of the collection of measure-distal transformations , 1996, Ergodic Theory and Dynamical Systems.

[13]  Terence Tao.  Szemerédi's regularity lemma revisited , 2006, Contributions Discret. Math..

[14]  H. Furstenberg.  Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions , 1977 .

[15]  E. Szemerédi.  On sets of integers containing k elements in arithmetic progression , 1975 .

[16]  Tamar Ziegler,et al.  Universal characteristic factors and Furstenberg averages , 2004, math/0403212.

[17]  K. F. Roth.  Irregularities of sequences relative to arithmetic progressions , 1967 .

[18]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

[19]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[20]  Paul Erdös,et al.  On Some Sequences of Integers , 1936 .

[21]  Jozef Skokan,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006 .

[22]  P. Varnavides,et al.  On Certain Sets of Positive Density , 1959 .

[23]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[24]  J. Spencer.  Ramsey Theory , 1990 .

[25]  Jean Bourgain,et al.  On Triples in Arithmetic Progression , 1999 .

[26]  Vojtech Rödl,et al.  Extremal problems on set systems , 2002, Random Struct. Algorithms.

[27]  Vitaly Bergelson,et al.  Set-polynomials and polynomial extension of the Hales-Jewett Theorem , 1999 .

[28]  E. Szemeri~di,et al.  On Sets of Integers Containing No Four Elements in Arithmetic Progression , .

[29]  D. Ornstein,et al.  The ergodic theoretical proof of Szemerédi's theorem , 1982 .

[30]  B. M. Fulk.  MATH , 1992 .

[31]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[32]  J. Girard.  Herbrand's Theorem and Proof-Theory , 1982 .

[33]  József Solymosi,et al.  A Note on a Question of Erdős and Graham , 2004, Combinatorics, Probability and Computing.

[34]  V. Rödl,et al.  The counting lemma for regular k-uniform hypergraphs , 2006 .

[35]  B. Host,et al.  ASPECTS OF UNIFORMITY IN RECURRENCE , 2000 .

[36]  K. F. Roth.  On Certain Sets of Integers , 1953 .

[37]  H. Furstenberg,et al.  An ergodic Szemerédi theorem for IP-systems and combinatorial theory , 1985 .

[38]  Vojtech Rödl,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006, Random Struct. Algorithms.

[39]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[40]  Bryna Kra,et al.  Nonconventional ergodic averages and nilmanifolds , 2005 .

[41]  J. Solymosi.  Note on a Generalization of Roth’s Theorem , 2003 .

[42]  Vojtech Rödl,et al.  The counting lemma for regular k‐uniform hypergraphs , 2006, Random Struct. Algorithms.

[43]  Mark Walters,et al.  Combinatorial Proofs of the Polynomial van der Waerden Theorem and the Polynomial Hales–Jewett Theorem , 2000 .

[44]  A SZEMERI DI TYPE THEOREM FOR SETS OF POSITIVE DENSITY IN R k , 2022 .

[45]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[46]  D. R. Heath-Brown.  Integer Sets Containing No Arithmetic Progressions , 1987 .

[47]  Mathias Schacht,et al.  Density theorems and extremal hypergraph problems , 2006 .