Factor-balancedness, linear recurrence, and factor complexity

TL;DR

The paper investigates factor-balancedness and its uniform variant for infinite words, introducing general $ ext{S}$-adic criteria that ensure (uniform) factor-balancedness for linearly recurrent words and applying them to Sturmian and ternary Arnoux--Rauzy words via bounded weak or partial quotients. It develops a comprehensive framework linking $ ext{S}$-adic representations, decisiveness, and substitutions to reduce factor-balancedness to letter-balancedness at appropriate levels, yielding a full characterization in key low-arity cases and shedding light on the balance-complexity trade-off. A major contribution is the analysis of factor-balancedness in relation to factor complexity, proving linear complexity for substitutive factor-balanced words and constructing Toeplitz-based examples of factor-balanced words with exponential complexity, thereby advancing understanding of the spectral/complexity landscape in symbolic dynamics. The results collectively clarify when balance properties imply stronger structural regularity and how such properties interact with dynamical spectrum and complexity, including explicit criteria for Sturmian and Arnoux--Rauzy systems and a concrete Toeplitz construction demonstrating positive entropy alongside factor-balance.

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, Berthé, Minervino, Steiner, and Thuswaldner on the relation between balancedness and discrete spectrum.

Figures (3)

• Figure 1: The Thue--Morse word ${\bf{t}}=01101001\cdots$ and its length-$2$ sliding block code $\pi({\bf{t}})=012023\cdots$.
• Figure 2: The Thue--Morse word ${\bf{t}}=01101001\cdots$ and its length-$2$ sum block code.
• Figure 3: A DFAO generating ${\bf{x}}$ in the Zeckendorf numeration system.

Theorems & Definitions (88)

• theorem 1.1: CDOPSS2025
• theorem 1.2
• proposition 1
• theorem 1.3
• theorem 1.4
• lemma 1: Durand-1999
• proposition 2: Ada03
• proposition 3
• lemma 2
• proof