Duke Mathematical Journal

On arithmetic structures in dense sets of integers

Ben Green

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We prove that if $A\subseteq\{1,\ldots N\}$ has density at least $(\log \log N)\sp {-c}$, where $c$ is an absolute constant, then $A$ contains a triple $(a, a+d,a+2d)$ with $d=x\sp 2+y\sp 2$ for some integers $x,y$, not both zero. We combine methods of T. Gowers and A. Sárközy with an application of Selberg's sieve. The result may be regarded as a step toward establishing a fully quantitative version of the polynomial Szemerédi theorem of V. Bergelson and A. Leibman.

Article information

Duke Math. J., Volume 114, Number 2 (2002), 215-238.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B25: Arithmetic progressions [See also 11N13]
Secondary: 11N36: Applications of sieve methods 11P55: Applications of the Hardy-Littlewood method [See also 11D85]


Green, Ben. On arithmetic structures in dense sets of integers. Duke Math. J. 114 (2002), no. 2, 215--238. doi:10.1215/S0012-7094-02-11422-7. https://projecteuclid.org/euclid.dmj/1087575409

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  • V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc. 9 (1996), 725--753.
  • H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204--256.
  • W. T. Gowers, A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529--551.
  • --. --. --. --., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001), 465--588., \CMP1 844 079; Erratum, Geom. Funct. Anal. 11 (2001), 869, \CMP1 866 805
  • B. Green, Notes on sieve theory: The Selberg sieve, http://dpmms.cam.ac.uk/~bjg23/expos.html
  • H. Halberstam and H.-E. Richert, Sieve Methods, London Math. Soc. Monogr. 4, Academic Press, London, 1974.
  • H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conf. Ser. in Math. 84, Amer. Math. Soc., Providence, 1994.
  • M. B. Nathanson, Elementary Methods in Number Theory, Grad. Texts in Math. 195, Springer, New York, 2000.
  • J. Pintz, W. L. Steiger, and E. Szemerédi, On sets of natural numbers whose difference set contains no squares, J. London Math. Soc. (2) 37 (1988), 219--231.
  • K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104--109.
  • I. Z. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), 205--209.
  • A. Sárközy, On difference sets of sequences of integers, I, Acta Math. Acad. Sci. Hungar. 31 (1978), 125--149.
  • --. --. --. --., On difference sets of sequences of integers, III, Acta Math. Acad. Sci. Hungar. 31 (1978), 355--386.
  • W. M. Schmidt, Small Fractional Parts of Polynomials, CBMS Regional Conf. Ser. in Math. 32, Amer. Math. Soc., Providence, 1977.
  • S. Srinivasan, On a result of Sárközy and Furstenberg, Nieuw Arch. Wisk. (4) 3 (1985), 275--280.
  • E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199--245.