On the 2-abelian complexity of generalized Cantor sequences

Highlights

• We prove that abelian complexity of 2-block coding of generalized Cantor sequences satisfying certain conditions is regular.

• We build a bridge between 2-abelian complexity of generalized Cantor sequences and abelian complexity of its 2-block coding.

• We prove that 2-abelian complexity of generalized Cantor sequences satisfying certain conditions is regular.

Abstract

In this paper, we study the generalized Cantor sequence c, which is an ℓ-automatic sequence. We prove that the abelian complexity of the 2-block sequence of c is ℓ-regular if the factor set of the sequence c is mirror invariant. As a consequence, we show that the 2-abelian complexity of a generalized Cantor sequence satisfying certain conditions is ℓ-regular.

Introduction

In recent years, abelian complexity of infinite sequences has been a widely studied topic. Early in the seventies of the last century, Coven and Hedlund [5] showed that the periodic and Sturmian words both can be characterized by using Parikh vectors. In 2011, Richomme, Saari and Zamboni [13] formally introduced abelian complexity through Parikh vectors. In the same paper, they characterized Thue-Morse minimal subshift by its factor complexity and abelian complexity simultaneously. Thereafter, the abelian complexity of some notable words, such as the regular paperfolding word and the Rudin-Shapiro word, were studied in [10] and [9] respectively. Madill and Rampersad [10] raised a question that whether the abelian complexity of any b-automatic sequence is always b-regular. In 2016, Blanchet-Sadri, Seita and Wise [3] proved that the abelian complexity of binary word w is ℓ-regular if w is a fixed point of arbitrary ℓ-uniform morphism φ satisfying that $||\phi (0){|}_0-|\phi (1){|}_0|\ne 1$.

The k-abelian complexity, as a generalization of abelian complexity, was firstly introduced in 1981 by Karhumäki [7]. The formal definition of the k-abelian complexity was given in 2013 by Karhumäki, Saarela and Zamboni [8]. Meanwhile, they computed the k-abelian complexity of all Sturmian words for every $k\ge 1$ and characterized ultimately periodic words by the k-abelian complexity. In 2014, Vandomme, Parreau and Rigo [12] conjectured that the 2-abelian complexity of the Thue-Morse word is 2-regular. An affirmative answer has been given independently by Greinecker [6] and by Parreau, Rigo, Rowland and Vandomme [11]. It is notable that in [11] they also proved that the 2-abelian complexity of the period-doubling word is 2-regular and raised the following general conjecture.

Conjecture 1

The 2-abelian complexity of any b-automatic sequence is b-regular.

For the general automatic sequence, its 2-abelian complexity is hard to investigate. In 2018, Chen, Lü and Wu [4] studied the k-abelian complexity of the Cantor sequence and proved that the k-abelian complexity of the Cantor sequence is always 3-regular for every $k\ge 1$. Here we will consider a class of automatic sequences which are generalizations of the Cantor sequence. First we give the definition. Let $k\ge 2$ be an integer and ${\{n_i\}}_{1\le i\le k-1}$ be $(k-1)$ positive integers. Set $\ell =k+{\sum }_{i=1}^{k-1}{n}_i$. The generalized Cantor sequence c is defined as the fixed point of the morphism $\sigma :1\mapsto {10}^{n_1}{10}^{n_2}1\cdots {10}^{n_{k-1}}1,0\mapsto 0^{\ell }$ beginning with 1, i.e.,

$$\mathbf{c}:={\sigma }^{\infty }(1)=c(0)c(1)\cdots ={10}^{n_1}{10}^{n_2}1\cdots {10}^{n_{k-1}}{10}^{\ell }\cdots .$$

Note that c is the Cantor sequence if $k=2$ and $n_1=1$.

For any infinite sequence w, we say the factor set of w is mirror-invariant if the reversal of u is also a factor of w whenever a finite word u is a factor of w. It is not hard to obtain that the factor set of the generalized Cantor sequence c is mirror-invariant if and only if $\sigma (1)$ is a palindrome, i.e., $\sigma (1)=\sigma {(1)}^R$ where $u^R$ denotes the reversal of u. By the definition of the morphism σ, $|\sigma (0){|}_0-|\sigma (1){|}_0=k\ge 2$. Following from [3] or Lemma 4 in Section 3, the abelian complexity of any generalized Cantor sequence c is ℓ-regular. In this paper, we focus on the 2-abelian complexity of c whose factor set is mirror-invariant. The first theorem is about the abelian complexity of the corresponding 2-block coding word $\mathbf{x}:=\mathrm{block}(\mathbf{c},2)$ (see Definition 4).

Theorem 1

If the factor set of the generalized Cantor sequence c is mirror-invariant, then the abelian complexity of the 2-block coding sequence x of c is ℓ-regular.

Before stating the second theorem, we need some notations. Let

$$\mathcal{A}:=\{i|0\le i\le \ell -1,c(i)=1\}=\{0\}\bigcup \{m+1+\sum _{j=1}^m{n}_j|1\le m\le k-1\}.$$

Let ${\mathcal{L}}_1:=\{\ell +j-i+1|i,j\in \mathcal{A}\text{and}i>j\}$ and ${\mathcal{L}}_2:=\{\ell +j-i+1|i,j\in \mathcal{A}\text{and}i\le j\}$. For any two sets $A,B\subset \mathbb{Z}$ and any two numbers $x,y\in \mathbb{Z}$, set

$$xA+yB:=\{xa+yb|a\in A,b\in B\},$$

and $xA-yB:=xA+(-y)B$. For convenience, write $xA-y:=xA-y\{1\}$. Now we have the following result about the 2-abelian complexity of c.

Theorem 2

If the factor set of the generalized Cantor sequence c is mirror invariant and ${\mathcal{L}}_1\cap ({\mathcal{L}}_2-\ell )=\varnothing$, then the 2-abelian complexity of c is ℓ-regular.

This paper is organized as follows. In Section 2, we will recall some basic definitions and notations. In Section 3, Theorem 1 will be proved. In the last section, we will prove Theorem 2.