Linear correlations amongst numbers represented by positive definite binary quadratic forms

Abstract

Abstract. Let f1,..., ft be positive definite binary quadratic forms, and letRfi(n) = |{(x, y) : fi(x, y) = n} | denote the corresponding representation functions. Employing methods developed by Green and Tao, we deduce asymptotics for linear correlations of these representation functions. More precisely, we study the expression En∈K∩[−N,N]d t∏ i=1 Rfi(ψi(n)), where the ψi form a system of affine linear forms no two of which are affinely related, and where K is a convex body. The minor arc analysis builds on the observation that polynomial subsequences of equidistributed nilsequences are still equidistributed, an observation that could be useful in treating the minor arcs of other arithmetic questions. As a very quick application we give asymptotics to the number of simultaneous zeros of certain systems of quadratic equations in 8 or more variables. Content