Indefinite quadratic polynomials of small signature

LetF()=Q()+L() be a real quadratic polynomial with no constant term. Suppose that the quadratic partQ() is indefinite of type (r, n-r). For an integerk4 we show that if min (r, n-r) >-k there exists a functionf (n, k)=–1/2+3/(4k+2)+O _k (1/n) with the following property. For any >0 and all large enoughX there is an integer vector 0 such that || X and.