Words with factor complexity $2n+1$ and minimal critical exponent

TL;DR

This paper confirms that the word with factor complexity 2n+1 has the minimal critical exponent, using a computer-assisted proof to verify the conjecture posed by Shallit and Shur.

Contribution

The paper proves, through computer-aided case analysis, that the word with factor complexity 2n+1 has the least possible critical exponent, confirming a conjecture by Shallit and Shur.

Findings

Confirmed the minimal critical exponent for words with factor complexity 2n+1.

Provided a computer-generated, human-readable proof of the conjecture.

Validated the specific critical exponent value as the lowest among such words.

Abstract

Word ${\mathbf G}$ is the fixed point of the morphism $\gamma=[01,2,02]$. In 2019, Shallit and Shur showed that ${\mathbf G}$ has factor complexity $2n+1$. They also showed that ${\mathbf G}$ has critical exponent $\mu=2+\frac{1}{\lambda^2-1}= 2.4808726\cdots$, where $\lambda=1.7548777$ is the real zero of $x^3-2x+x-1=0$. They conjectured that this was the least possible critical exponent among words with factor complexity $2n+1$. We confirm their conjecture. The proof, using an intricate case analysis, is by computer. The relevant program generates a `human readable' proof.