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

James D. Currie

Abstract

Word ${\mathbf G}$ is the fixed point of the morphism $γ=[01,2,02]$. In 2019, Shallit and Shur showed that ${\mathbf G}$ has factor complexity $2n+1$. They also showed that ${\mathbf G}$ has critical exponent $μ=2+\frac{1}{λ^2-1}= 2.4808726\cdots$, where $λ=1.7548777$ 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.

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

Abstract

Word is the fixed point of the morphism . In 2019, Shallit and Shur showed that has factor complexity . They also showed that has critical exponent , where is the real zero of . They conjectured that this was the least possible critical exponent among words with factor complexity . We confirm their conjecture. The proof, using an intricate case analysis, is by computer. The relevant program generates a `human readable' proof.

Paper Structure

This paper contains 7 sections, 14 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Word preliminaries
  3. Main Theorem
  4. Eligible words and their factors
  5. Proof of the structure theorem
  6. Proofs of Lemmas
  7. A sharpened structure theorem and open problems

Key Result

Theorem 1

mol1 Let ${\mathbf w}\in\Sigma_2^\omega$ be a $14/5$-power-free rich word. For every $n\geq 1$, a suffix of ${\mathbf w}$ has the form $f(\phi^n(w_n))$ or $f(g(\phi^n(w_n)))$ for some word $w_n\in\Sigma_3^\omega$ where

Figures (2)

  • Figure 1: Potential morphisms $h$ such that some eligible word contains $h(g)$ for each factor $g$ of ${\mathbf G}$.
  • Figure 2: Potential morphisms $h$ such that some eligible word contains $h(g)$ for each factor $g$ of ${\mathbf G}'$.

Theorems & Definitions (28)

  • Theorem 1
  • Lemma 1: König's lemma
  • Theorem 2: Main Theorem
  • Theorem 3: Structure Theorem
  • Lemma 2
  • Remark 1
  • proof : Proof of main theorem
  • Lemma 3
  • Lemma 4: Morphism Lemma
  • Lemma 5: Dual Morphism Lemma
  • ...and 18 more