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New bounds for Szemerédi’s theorem. II: A new bound for \(r_4(N)\). (English) Zbl 1158.11007
Chen, W. W. L. (ed.) et al., Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press (ISBN 978-0-521-51538-2/hbk). 180-204 (2009).
The paper under review considers the celebrated theorem of Szemerédi for progressions of length 4. Specifically we write \(r_4(N)\) for the size of the largest subset of \(\{1,\dots,N\}\) which contains no non-trivial four term arithmetic progressions (a trivial progression is one with common difference 0). E. Szemerédi [Acta Math. Acad. Sci. Hung. 20, 89–104 (1969; Zbl 0175.04301)] provided a proof that \(r_4(N)=o(N)\) in 1969 and a few years later K. F. Roth [J. Lond. Math. Soc. 28, 104–109 (1953; Zbl 0050.04002)] provided a different proof of this in an effort to improve bounds. It was a major breakthrough in 1998 when W. T. Gowers [Geom. Funct. Anal. 8, No. 3, 529–551 (1998; Zbl 0907.11005)] showed that \[ r_4(N) = O(N\exp(-\Omega(\log \log \log N))). \] It is the purpose of the paper at hand to improve this to \[ r_4(N)=O(N \exp(-\Omega(\sqrt{\log \log N}))), \] and in the forthcoming part III of the series of papers the authors remove the square root although doing so considerably complicates the argument.
All proofs of Szemerédi’s theorem (for any length of progression) proceed by establishing a dichotomy: either a set ‘behaves randomly’ in which case it has the desired number of progressions, or it correlates with some sort of structure, meaning that it has increased density on a structured object e.g. a long arithmetic progression.
In the case of progressions of length \(3\) ‘behaving randomly’ means having no large Fourier coefficients. Now it turns out that if the number of three term progressions is not about what it is for a random set then there is not just one large Fourier coefficient but many and one may try incrementing the density simultaneous for all these large Fourier coefficients. Such a strategy was developed by Szemerédi and Heath-Brown to improve the bounds in the length \(3\) case and the authors effectively adapt this to the \(4\)-term case in the present paper.
The argument is involved, but far less so than the sequel (essentially because one may linearize more frequently) and so is a natural intermediate paper.
Part I, see Proc. Lond. Math. Soc. (3) 98, No. 2, 365–392 (2009; Zbl 1180.11006).
For the entire collection see [Zbl 1155.11004].

11B25 Arithmetic progressions
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