On arithmetic structures in dense sets of integers
The main aim of the paper is to prove the result that if \(A\subset\{1,2,\dots,N\}\) has density at least \((\log\log N)^{-c}\), where \(c\) is an absolute constant, then \(A\) contains a \(3\)-term arithmetic progression whose common difference is positive and of the form \(x^2+y^2\) for some integers \(x,y\) not both zero. The proof method is based on ideas developed by Gowers and Sárközy combined with an application of the Selberg sieve. Using a new variant of Sárközy's proof, the author also shows that there is a constant \(c\) such that, if \(A\subset\{1,2,\dots,N\}\) has density at least \(c(\log\log N)^{-1/11}\), then \(A\) contains two elements \(a,a^\prime\) with \(a-a^\prime\) a nonzero square.
scientific article; zbMATH DE number 1820923