The Repetition Threshold for Rote Sequences

TL;DR

This work determines the repetition threshold for Rote sequences, binary words with factor complexity $\rho_{\mathbf x}(n)=2n$, proving the threshold is $\frac{5}{2}$. The authors combine a finite-exhaustive lower bound with a constructive infinite word $\mathbf q$ and rigorous automated verification via the Walnut theorem prover, aided by a novel morphism-based automata-generation technique that yields a compact numeration system. The central construction shows $\mathbf q$ is a Rote word with $\mathrm{ce}(\mathbf q)=\frac{5}{2}$ by proving $\mathbf q$ is $(\frac{5}{2})^+$-power-free and $\rho_{\mathbf q}(n)=2n$, and by identifying the unique $\frac{5}{2}$-exponent factor $1001100110$. Overall, the paper advances the understanding of repetition phenomena in Rote words and demonstrates a scalable computational framework for proving combinatorial properties of automatic and morphic sequences.

Abstract

We consider Rote words, which are infinite binary words with factor complexity $2n$. We prove that the repetition threshold for this class is $5/2$. Our technique is purely computational, using the Walnut theorem prover and a new technique for generating automata from morphisms due to the first author and his co-authors.

Figures (2)

• Figure 1: Addressing and sequence automata for ${h(a) = ab, h(b) = cb, h(c) = a}$
• Figure 2: Addressing and sequence automata for ${\bf q}$

Theorems & Definitions (10)

• Proposition 1
• proof
• Theorem 2
• Theorem 3
• proof
• proof
• Remark 4
• Remark 5
• Remark 6
• Remark 7