Squarefree words with interior disposable factors

TL;DR

This paper constructs an infinite ternary squarefree word with interior factors that can be deleted without losing squarefreeness and characterizes deletion positions in Thue's squarefree word, advancing understanding of squarefree words.

Contribution

It provides a partial solution to Harju's problem by constructing such a word and analyzes deletion properties in Thue's classic squarefree word.

Findings

Existence of an infinite ternary squarefree word with interior disposable factors for large k

Characterization of deletion positions in Thue's squarefree word

Insights into the structure of squarefree words and their resilience to symbol deletion

Abstract

We give a partial answer to a problem of Harju by constructing an infinite ternary squarefree word $w$ with the property that for every $k \geq 3312$ there is an interior length-$k$ factor of $w$ that can be deleted while still preserving squarefreeness. We also examine Thue's famous squarefree word (generated by iterating the map $0 \to 012$, $1 \to 02$, $2 \to 1$) and characterize the positions $i$ for which deleting the symbol appearing at position $i$ preserves squarefreeness.