Critical exponents of infinite balanced words

TL;DR

This paper constructs infinite balanced words with specific critical exponents over small alphabets and explores computational methods and theorem proving techniques for analyzing their properties.

Contribution

It introduces explicit constructions of balanced words with particular critical exponents and demonstrates the use of automated theorem proving for their analysis.

Findings

Constructed balanced words over 3 and 4-letter alphabets with specified critical exponents.

Provided computational candidates for balanced words with small critical exponents over larger alphabets.

Applied the Walnut theorem prover to verify properties of these words.

Abstract

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated theorem prover Walnut.