# On the $k$-abelian complexity of the Cantor sequence

## TL;DR

This paper proves that for all integers k ≥ 1, the k-abelian complexity of the Cantor sequence is a 3-regular sequence, revealing a structured regularity in its combinatorial complexity.

## Contribution

It establishes that the k-abelian complexity function of the Cantor sequence is 3-regular for all k ≥ 1, a novel result in combinatorics on words.

## Key findings

- k-abelian complexity is 3-regular for the Cantor sequence
- The result applies to all integers k ≥ 1
- Provides insight into the structure of the Cantor sequence's complexity

## Abstract

In this paper, we prove that for every integer $k \geq 1$, the $k$-abelian complexity function of the Cantor sequence $\mathbf{c} = 101000101\cdots$ is a $3$-regular sequence.