  In the present paper, we study the geometric discrepancy with respect to
families of rotated rectangles. The well-known extremal cases are the
axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in
all possible directions (polynomial discrepancy). We study several intermediate
situations: lacunary sequences of directions, lacunary sets of finite order,
and sets with small Minkowski dimension. In each of these cases, extensions of
a lemma due to Davenport allow us to construct appropriate rotations of the
integer lattice which yield small discrepancy.
