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 \({x\in {\Sigma }_m:\forall k, x_kx_{2k}…x_{nk}=0}\), where \(n\geq 3\). We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.