On arithmetic structures in dense sets of integers
Green, Ben
Duke Math. J., Tome 115 (2002) no. 1, p. 215-238 / Harvested from Project Euclid
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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.
Publié le : 2002-08-15
Classification:  11B25,  11N36,  11P55 
@article{1087575409,
     author = {Green, Ben},
     title = {On arithmetic structures in dense sets of integers},
     journal = {Duke Math. J.},
     volume = {115},
     number = {1},
     year = {2002},
     pages = { 215-238},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1087575409}
}

Green, Ben. On arithmetic structures in dense sets of integers. Duke Math. J., Tome 115 (2002) no. 1, pp.  215-238. http://gdmltest.u-ga.fr/item/1087575409/