POSITIVE VALUES OF NONHOMOGENEOUS INDEFINITE QUADRATIC-FORMS OF TYPE (3, 2)

Let GAMMA(r,n-r) denote the infimum of all numbers GAMMA > 0 such that for any real indefinite quadratic form Q in n variables of type (r, n-r), determinant D not-equal 0 and real numbers c1,...,c(n), there exist (x1,...,x(n)) = (c1,...,c(n)) (mod 1) satisfying 0 < Q(x1,...,x(n)) less-than-or-equal-to (GAMMA absolute value of D)1/n. It is shown that the value GAMMA3,2 = 16 is isolated and that the second minimum is 8. We also determine all the forms for which the constants 16 and.8 are required.