On arithmetic structures in dense sets of integers

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.