Approximating numbers with missing digits by algebraic numbers

We show that for a given base b and a proper subset E subset of {0,...,b - 1}, #E < b - 1, the set of numbers x E [0, 1] that have no digits from E in their expansion to base b consists almost exclusively of S*-numbers of type at most min{2, log b/log(b - #E)}. We also give upper bounds on the Hausdorff dimension of some exceptional sets.