Factor-balancedness, linear recurrence, and factor complexity

2026-02-03

AI Summary

This study investigates the relationship between factor balance and factor complexity in infinite words and explores their connection to discrete spectrum. By leveraging S-adic representations, linearly recurrent structures, substitution systems, and combinatorics on words, the authors establish sufficient conditions for linearly recurrent infinite words to satisfy (uniform) factor balance. The main contributions include a generalization of Adamczewski’s results on primitive substitutive words, the first construction of factor-balanced words with exponential factor complexity, and a complete characterization of uniform factor balance for Sturmian words and ternary Arnoux–Rauzy words with bounded weak partial quotients. Furthermore, the work introduces a general criterion for verifying uniform factor balance and constructs pivotal counterexamples that deepen the understanding of the interplay between balance properties and discrete spectrum.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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

In the study of infinite words, various notions of balancedness provide quantitative measures for how regularly letters or factors occur, and they find applications in several areas of mathematics and theoretical computer science. In this paper, we study factor-balancedness and uniform factor-balancedness, making two main contributions. First, we establish general sufficient conditions for an infinite word to be (uniformly) factor-balanced, applicable in particular to any given linearly recurrent word. These conditions are formulated in terms of $\mathcal{S}$-adic representations and generalize results of Adamczewski on primitive substitutive words, which show that balancedness of length-2 factors already implies uniform factor-balancedness. As an application of our criteria, we characterize the Sturmian words and ternary Arnoux--Rauzy words that are uniformly factor-balanced as precisely those with bounded weak partial quotients. Our second main contribution is a study of the relationship between factor-balancedness and factor complexity. In particular, we analyze the non-primitive substitutive case and construct an example of a factor-balanced word with exponential factor complexity, thereby making progress on a question raised in 2025 by Arnoux, Berth\'e, Minervino, Steiner, and Thuswaldner on the relation between balancedness and discrete spectrum.

Problem

• factor-balancedness
• factor complexity
• linear recurrence
• infinite words
• balancedness

Innovation

• factor-balancedness
• linear recurrence
• factor complexity
• S-adic representation
• substitutive words