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arXiv · 2507.09387

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

James D. Currie
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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.

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James D. Currie. 2025-07-12. Words with factor complexity $2n+1$ and minimal critical exponent. https://arxiv.org/abs/2507.09387

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