In 1969, William Veech introduced two subsets K_1(*θ*) and K_0(*θ*) of R/Z which are defined in terms of the continued fraction expansion of *θ*. These subsets are known to give information about the dynamics of certain skew products of the unit circle. We show that the Hausdorff dimension of K_i(*θ*) can achieve any value between zero and one.