Small zeros of quadratic forms modulo p, III

Let Q(x)=Q(x1, x2, …, xn) be a quadratic form over ℤ and p be an odd prime. Let ‖x‖=max|xi|. We show that for n≥4, there exists a nonzero solution of the congruence Q(x)э0 mod p with ‖x‖≪p1/2, which is best possible. A similar result is proven for boxes not centered at the origin