Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Sigma(m) ={0, ... , m-1}(N) that are invariant under multiplication by integers. The results apply to the sets {x is an element of Sigma(m): for all k, x(k)x(2k) ... x(nk) = 0}, where n >= 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.