Properties of a Class of Toeplitz Words

TL;DR

This paper investigates Stewart words, a class of Toeplitz words, analyzing their critical exponent, pattern avoidance, palindromic factors, and subword complexity using automata-theoretic methods.

Contribution

It introduces automata-based techniques to analyze Stewart words and determines key combinatorial properties that were previously unknown.

Findings

Stewart words avoid the pattern xxyyxx.

The critical exponent of Stewart words is determined.

All palindromic factors of Stewart words are characterized.

Abstract

We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form $\alpha\beta\gamma$, where $\alpha,\beta,\gamma$ is any permutation of the symbols 0,1,?. We determine the critical exponent of the Stewart words, prove that they avoid the pattern $xxyyxx$, find all factors that are palindromes, and determine their subword complexity. An interesting aspect of our work is that we use automata-theoretic methods and a decision procedure for automata to carry out the proofs.