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Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms
Authors:Wolfgang Müller
Abstract:Let Q 1,…,Q r be quadratic forms with real coefficients. We prove that the set {(Q1(x),?,Qr(x)) | x ? Bbb Zs}{(Q_1(x),ldots ,Q_r(x)),vert, xin{Bbb Z}^s} is dense in Bbb Rr{Bbb R}^r , provided that the system Q 1(x) = 0,…,Q r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q 1,…,Q r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form.
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