On the value distribution of positive definite quadratic forms (Q621773)

Let \(Q\) be a positive definite quadratic form in \(k\) variables, with \(k\geq2\), and let \(\lambda_0, \lambda_1, \lambda_2,\dots\) denote the infinite sequence obtained by arranging the values of \(Q(x)\) at integer points \(x\in{\mathbb Z}^k\), with multiplicity, in ascending order, so that \(0=\lambda_0<\lambda_1\leq\lambda_2\leq\cdots\). Let \(l\) be an integer \(\geq2\), and for an \((l-1)\)-dimensional interval \(I=\prod_{j=2}^l I_j\), define the \textit{\(l\)-level correlation function} \(K_I^{(l)}(R)\) to be the number of \(l\)-tuples of suffixes \((i_1,\dots,i_l)\) such that \(\lambda_{i_j}\leq R\) \((1\leq j\leq l)\) and \(\lambda_{i_j}-\lambda_{i_1}\in I_j\) \((2\leq j\leq l)\). The author proved in his previous paper [``Systems of quadratic diophantine inequalities and the value distribution of quadratic forms'', Monatsh. Math. 153, No. 3, 233--250 (2008; Zbl 1184.11011)] that when \(k\geq5\) and \(2\leq l< k/4+1\), one has \[ K_I^{(l)}(R)\sim (kD_k)^l 2^{1-l}(l(k-2)+2)^{-1}\text{vol}(I)R^{lk-2(l-1)} \quad \text{(as \(R\rightarrow\infty\)),} \] where \(D_k\) denotes the volume of the ellipsoid defined by \(Q(x)\leq1\). In the present paper, he establishes the latter formula for \(k\geq4\) and all \(l\geq2\), and also for \(k=3\) and \(l=2\), 3. The proof is based on a variant of the Davenport-Heilbronn circle method, and it is pointed out that in a certain sense concretely described in the paper, the latter restrictions on \(k\) and \(l\) are the best one can expect by the method.