Low complexity binary words avoiding $(5/2)^+$-powers

TL;DR

This work gives a precise, Restivo‑Salemi–style structure theorem for binary Rote words that avoid $(5/2)^+$‑powers, confirming a conjecture of Ollinger and Shallit. The authors develop a two‑tier description: (i) a First Structure Theorem showing that final segments of proper words are obtained via the morphisms $f$ (and $h$ for antiproper words), and (ii) a Second Structure Theorem classifying Rote words by their length‑4 factor sets and proving that final segments are of the form $g^*(f^n(oldsymbol{u}))$ (and variants with complements/reversals) where $oldsymbol{u}$ is proper. The approach combines careful combinatorial analysis of 4‑letter factors with recursive morphic constructions and is complemented by computational backtracking that validates the imposed length bounds. The results provide a complete, recursive description of the low‑complexity, $(5/2)^+$‑power‑free Rote words, linking their structure to explicit morphisms and offering a template for analogous results in related power‑free word classes.

Abstract

Rote words are infinite words that contain $2n$ factors of length $n$ for every $n \geq 1$. Shallit and Shur, as well as Ollinger and Shallit, showed that there are Rote words that avoid $(5/2)^+$-powers and that this is best possible. In this note we give a structure theorem for the Rote words that avoid $(5/2)^+$-powers, confirming a conjecture of Ollinger and Shallit.

Theorems & Definitions (23)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3