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

TL;DR

This paper confirms two conjectures by J. Shallit regarding the factor complexity and critical exponent of a class of infinite words including Thue-Morse and related sequences, for all positive integers k.

Contribution

It proves that the two conjectures hold for the entire class of sequences $x_k$, extending Shallit's initial results for specific cases to all positive integers.

Findings

Both conjectures are valid for all $x_k$ sequences.

The factor complexity of $x_k$ sequences is characterized.

The critical exponent of $x_k$ sequences is determined.

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 $k$.