# Thue-Morse-Sturmian words and critical bases for ternary alphabets

## TL;DR

This paper investigates the structure of unique beta-expansions over three-letter alphabets using S-adic words, extending known thresholds for binary alphabets to more complex ternary cases.

## Contribution

It introduces a new generalization of the Komornik-Loreti constant for three-letter alphabets using Thue-Morse and Sturmian words.

## Key findings

- Determined the value of the generalized Komornik-Loreti constant for ternary alphabets.
- Connected S-adic words with thresholds for unique expansions.
- Extended binary alphabet results to ternary cases.

## Abstract

The set of unique $\beta$-expansions over the alphabet $\{0,1\}$ is trivial for $\beta$ below the golden ratio and uncountable above the Komornik-Loreti constant. Generalisations of these thresholds for three-letter alphabets were studied by Komornik, Lai and Pedicini (2011, 2017). We use S-adic words including the Thue-Morse word (which defines the Komornik-Loreti constant) and Sturmian words (which characterise generalised golden ratios) to determine the value of a certain generalisation of the Komornik-Loreti constant to three-letter alphabets.