Lattice points and generalized diophantine conditions

We prove that (integral (2 pi)0 (\N-theta(t) - \D \ t(2))(2) d theta)(1 2 = ) O(t(2/3)), where D-theta is a rotation of a convex domain in R-2 and N-th eta(t) = not equal {Z(2) boolean AND tD(theta)}. It follows that for any de lta > 0, there exists a set of 0's of measure 2 pi - delta, such that for t epsilon Lambda, where Lambda is any lacunary sequence. \N-theta(t) - \D \ t(2)\ less than or equal to C(Lambda)t(2/3) log(t). Moreover, we prove, und er some additional assumptions, that for almost every theta, N-theta(t) - \D \ t(2) = O(t(2 3)), (*) up to a small logarithmic transgression. We also prove that if D is convex and finite type, and also in some infinit e type situations. N-tau (t) = not equal {Z(2) boolean AND tD + tau}, tau e psilon T-2. the two-dimensional torus, and parallel toN(tau)(t) - t(2) \D \ parallel to (L2(T)2) less than or equal to Ct(1/2), (**) the optimal bound. then N-(0,N-0) (t) = t(2) \D \ + O(t(2/3)). We conclude that (**) cannot in general hold if the boundary of D has order of contact greater than or equal to4 with one of its tangent lines. (C) 2001 Academic Press.