Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set is dense in , 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. Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria