Difference of Cantor sets and frequencies in Thue--Morse type sequences

TL;DR

This paper extends the study of Cantor set intersections to more general cases and introduces a practical method to determine digit frequencies in Thue--Morse sequences, linking fractal geometry with combinatorial sequence analysis.

Contribution

It generalizes previous results on Cantor sets and develops a new method for analyzing digit frequencies in Thue--Morse type sequences.

Findings

Extended Hausdorff dimension results to broader Cantor sets

Developed a practical frequency determination method for finite blocks

Linked digit frequencies with fractal intersection properties

Abstract

In a recent paper, Baker and Kong have studied the Hausdorff dimension of the intersection of Cantor sets with their translations. We extend their results to more general Cantor sets. The proofs rely on the frequencies of digits in unique expansions in non-integer bases. In relation with this, we introduce a practical method to determine the frequency of any given finite block in Thue--Morse type sequences.