Transition Property For Cube-Free Words

TL;DR

This paper establishes a transition property for cube-free words over arbitrary alphabets, enabling solutions to longstanding open problems and revealing new insights into the structure and complexity of such words.

Contribution

It proves a transition property for cube-free words over any finite alphabet, including the binary case, and applies this to solve three open problems and analyze word complexity.

Findings

Existence of a transition word connecting infinite extensions of cube-free words.

Resolution of three open problems by Restivo and Salemi.

Infinite cube-free words with very high subword complexity.

Abstract

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair $(u,v)$ of $d$-ary cube-free words, if $u$ can be infinitely extended to the right and $v$ can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word $w$ over the same alphabet such that $uwv$ is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained "transition property", together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.