Binary Patterns in Binary Cube-Free Words: Avoidability and Growth

TL;DR

This paper investigates the avoidability of binary patterns in binary cube-free words, establishing bounds, growth rates, and providing new examples of pattern-avoiding languages with various growth behaviors.

Contribution

It determines the exact boundary between avoidable and unavoidable patterns, shows all avoidable patterns are D0L-avoidable, and introduces a polynomial growth pattern-avoiding language.

Findings

All avoidable patterns are D0L-avoidable.

Most avoiding languages have exponential growth, except overlap-free.

A new polynomial growth pattern-avoiding language is presented.

Abstract

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.