Constructing Premaximal Binary Cube-free Words of Any Level

TL;DR

This paper investigates the structure of binary cube-free words, focusing on those that cannot be extended infinitely while maintaining cube-freeness, and demonstrates the existence of such words with arbitrarily long finite extensions.

Contribution

It introduces the concept of premaximal binary cube-free words and proves their existence for any extension length, advancing understanding of cube-free word boundaries.

Findings

Existence of premaximal binary cube-free words with arbitrary extension lengths

Construction methods for such words in binary alphabet

Insights into the structure and limitations of cube-free languages

Abstract

We study the structure of the language of binary cube-free words. Namely, we are interested in the cube-free words that cannot be infinitely extended preserving cube-freeness. We show the existence of such words with arbitrarily long finite extensions, both to one side and to both sides.