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

Let Q(1), ..., Qr be quadratic forms with real coefficients. We prove that the set {(Q(1)(x), ... ,Q(r)(x)) vertical bar x is an element of Z(s)} is dense in 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.