Lattice points in rotated convex domains

If B subset of R-d (d >= 2) is a compact convex domain with a smooth boundary of finite type, we prove that for almost every rotation theta is an element of SO(d) the remainder of the lattice point problem, P-theta B (t) is of order O-theta(t(d-2+2/(d+1)-zeta d)) with a positive number zeta(d). Furthermore we extend the estimate of the above type, in the planar case, to general compact convex domains.