Duke Mathematical Journal

On arithmetic structures in dense sets of integers

Ben Green

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Abstract

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

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

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575409

Digital Object Identifier
doi:10.1215/S0012-7094-02-11422-7

Mathematical Reviews number (MathSciNet)
MR1920188

Zentralblatt MATH identifier
1020.11010

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

Citation

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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References

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