Factor-balancedness, linear recurrence, and factor complexity

TL;DR

This paper investigates conditions under which infinite words are factor-balanced and explores the relationship between factor-balancedness and factor complexity, providing new characterizations and examples in the context of linearly recurrent words.

Contribution

It establishes general criteria for factor-balancedness using $ ext{S}$-adic representations and characterizes uniformly factor-balanced Sturmian and Arnoux--Rauzy words, also analyzing the link with factor complexity.

Findings

Criteria for factor-balancedness in linearly recurrent words.

Characterization of uniformly factor-balanced Sturmian and Arnoux--Rauzy words.

Example of a factor-balanced word with exponential factor complexity.

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.