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 {(Q₁(x),?,Q_r(x)) | x ? \Bbb Z^s}\{(Q_1(x),\ldots ,Q_r(x))\,\vert\, x\in{\Bbb Z}^s\} is dense in \Bbb R^r{\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.