On two conjectures of Shallit about Thue-Morse-like sequences

TL;DR

We address the problem of understanding factor complexity and repetition properties of the Thue-Morse-like sequences $x_k$ obtained as the projection of the fixed point $\mathbf{u}_k=\xi_k^\omega(0)$ of the Narayana-type morphism $\xi_k$. The authors develop a detailed bispecial-factor analysis via Klouda's triplet framework, establish overlap-free-ness of $\mathbf{u}_k$, and study the projection $\pi_k$ to $x_k$ to transfer structural information. They prove, for all $k\ge1$, that the first difference of factor complexity satisfies $\Delta\mathcal{C}(N)\in\{4k-2,4k\}$ for large $N$, and that the critical exponent of $x_k$ is $k+1$ with asymptotic exponent $2$, with the extremal words $0^{k+1}$ and $1^{k+1}$ attaining the bound. These results extend the known cases $k=1,2,3$ to all $k$ and provide a complete combinatorial description via bispecial factors and return words, illustrating a language invariant under letter exchange. The findings have implications for the theory of morphic words, complexity bounds, and repetitions in binary Thue-Morse-like sequences.

Abstract

We study a class of infinite words $x_k$ , where $k$ is a positive integer, recently introduced by J. Shallit. This class includes the Thue-Morse sequence $x_1$, the Fibonacci-Thue-Morse sequence $x_2$, and the Allouche-Johnson sequence $x_3$. Shallit stated and for $k = 3$ proved two conjectures on properties of $x_k$. The first conjecture concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every $k$.

Figures (1)

• Figure 1: All left special factors of $\mathbf{x}_k$, where $k=3$, of length $N$ with $10\leq N \leq 25$, are prefixes of the depicted factors or their twins.

Theorems & Definitions (27)

• Conjecture 1: Sha25, Conjecture 45
• Conjecture 2: Sha25, Conjecture 44
• Theorem 3: DolceDP2023, Theorem 3
• Theorem 6
• Proposition 7
• proof
• Lemma 8
• proof
• Lemma 9
• proof