The Oppenheim conjecture, proved by Margulis [Mar1] (see also [Mar2]), asserts that for a nondegenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the set Q(Zn ) is dense. In [EMM] (where we used the lower bounds established in [DM]) a quantitative version of the conjecture was established. Namely: Let ρ be a continuous positive function on the sphere {v ∈ Rn | v = 1}, and let Ω = {v ∈ Rn | v