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

TL;DR

This paper investigates the fractal dimensions of certain invariant sets in symbolic spaces under integer multiplication, revealing that Hausdorff and Minkowski dimensions often differ for these sets.

Contribution

It provides explicit calculations of Hausdorff and Minkowski dimensions for invariant symbolic sets under multiplication, highlighting their typical differences.

Findings

Hausdorff and Minkowski dimensions often differ for these sets

Explicit formulas for dimensions of invariant sets under multiplication

Application to sets defined by digit restrictions in symbolic spaces

Abstract

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 $\{x\in \Sigma_m: \forall\, k, \ x_k x_{2k}... x_{n k}=0\}$, where $n\ge 3$. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.