Linear correlations amongst numbers represented by positive definite binary quadratic forms (Q2896950)

Let \(f_1,\ldots,f_t\) be primitive, positive definite, binary quadratic forms, and for each define the function NEWLINE\[NEWLINE R_{f_i}(n) := \#\{(x,y) \in \mathbb{Z}^{2} : f(x,y)=n\}. NEWLINE\]NEWLINE In this paper, the author's main result is an asymptotic formula for certain correlations of such functions. More precisely, for any fixed natural numbers \(d,t\), any convex body \(K \subseteq [-N,N]^{d}\), and any affine-linear forms \(\Psi_{i}\) on \(\mathbb{Z}^{d}\) with bounded coefficients, such that \(\Psi_i(K) \subseteq [1,N]\) for all \(i\), and any two of which are linearly independent, it is shown that NEWLINE\[NEWLINE \sum_{n \in K \cap \mathbb{Z}^{d}} R_{f_1}(\Psi_1(n)) \cdots R_{f_t}(\Psi_t(n)) = \text{vol}(K)\, \beta'_{\infty} \prod_p \beta_p + o(N^d). NEWLINE\]NEWLINE Here the \(\beta_{p}\) and \(\beta'_{\infty}\) are certain explicit factors.NEWLINENEWLINEAs the author explains, the main result was already known (with stronger error term) for certain \(d\) and \(t\), particularly in the sums of two squares case where \(f_{i}(x,y) := x^{2}+y^{2}\) for all \(i\). The significance of the author's work is that there is no restriction on \(d\) and \(t\), apart from the underlying assumption that any two of the \(\Psi_{i}\) are linearly independent. This assumption excludes certain very interesting and difficult cases, but the author can nevertheless treat many important cases that were inaccessible before. For example, Theorem 1.3 of the present paper is an application to counting simultaneous solutions of certain systems of \(t-2\) forms in \(2t\) variables.NEWLINENEWLINEThe proof of the author's main theorem is based on the ``nilpotent Hardy-Littlewood method'', (that is to say, the methods of Green and Tao), as in the author's previous paper [Proc. Lond. Math. Soc. (3) 104, No. 4, 827--858 (2012; Zbl 1294.11168)] where the functions \(R_{f_i}(n)\) were replaced by the divisor function \(\tau(n) := \sum_{d\mid n} 1\). One of the main tasks is to construct a pseudorandom majorant function \(\nu(n)\) that (roughly speaking) is a good upper bound for all the functions \(R_{f_i}(n)\), and that can be shown to satisfy two conditions called the \textit{linear forms condition} and the \textit{correlation condition}. This work takes up about half the paper. As the author explains, one can upper bound \(R_{f_i}(n)\) by certain divisor sums \(\sum_{d\mid n} \left(\frac{D}{d}\right)\) involving the Kronecker symbol, and one can think of such sums as behaving roughly like the divisor function \(\tau(n)\) up to certain conditions on the prime factors of \(n\). Thus \(\nu(n)\) is constructed by combining a majorant for \(\tau(n)\) (from the author's previous paper) with a sieve theory part that efficiently encodes the prime factor conditions.NEWLINENEWLINEHaving constructed \(\nu(n)\), the rest of the paper is an application of the machinery of \textit{B. Green} and \textit{T. Tao} [Ann. Math. (2) 171, No. 3, 1753--1850 (2010; Zbl 1242.11071)], which takes the place of the pointwise major and minor arc analysis in the classical Hardy-Littlewood circle method. This is again somewhat similar to the author's previous work, but the arguments are now considerably more complicated because of the degree two polynomials in the definition of \(R_{f_i}(n)\).NEWLINENEWLINEThe quality of exposition in this paper is very high, but the final parts are also very technical. Thus a novice reader might do well to first look at the author's previous work [loc. cit.], which is simpler in various respects.