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

Abstract

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space ${\Sigma }_m=\{0,...,m-1{\}}^{\mathbb{N}}$ that are invariant under multiplication by integers. The results apply to the sets $\{x\in {\Sigma }_m:\forall k,x_k{x}_{2k}...x_{nk}=0\}$, where $n\ge 3$. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

arXiv comment22 pages

References in corpus (1)

• Level sets of multiple ergodic averages