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

2025-06-23

AI Summary

This paper addresses the low-complexity construction of binary infinite words avoiding $(5/2)^+$-powers, specifically within the class of Rote words—i.e., binary infinite words with factor complexity exactly $2n$. Building on a structural conjecture by Ollinger and Shallit concerning such Rote words, we provide the first rigorous proof via combinatorial language analysis, fine-grained characterization of factor complexity, and equivalence-class construction, augmented by automata-theoretic and formal language techniques. Our main results are: (i) all Rote words avoiding $(5/2)^+$-powers arise precisely from a specific bijective coding whose underlying structure is fully determined by a variant of the Thue–Morse sequence; and (ii) $(5/2)^+$ is a tight threshold—such words exist, yet no Rote word avoids $(5/2 - varepsilon)^+$-powers for any $varepsilon > 0$. This resolves the long-standing factor complexity boundary problem for power avoidance and establishes a foundational bridge in low-complexity avoidability theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Characterize Rote words avoiding (5/2)^+-powers
• Prove structure theorem for such low-complexity words
• Confirm Ollinger-Shallit conjecture on power avoidance

Innovation

• Structure theorem for Rote words
• Avoiding (5/2)^+-powers
• Confirming Ollinger-Shallit conjecture