Factor-balancedness, linear recurrence, and factor complexity

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.

💡 Research Summary

The paper investigates two closely related regularity properties of infinite words: factor‑balancedness (the bounded discrepancy of occurrences of any finite factor across equal‑length blocks) and its stronger version, uniform factor‑balancedness (where the bound is independent of the blocks). While letter‑balancedness has been extensively studied, factor‑balancedness depends on finer combinatorial structure and has received far less attention.

The authors first develop a general framework based on S‑adic representations. They introduce three technical conditions—denoted (P), (F) and (D)—that a “congenial sequence” ((\tau_n,a_n){n\ge0}) must satisfy. For a linearly recurrent word (x) generated by such a sequence, let (x^{(n)}) be the limit of the iterated morphisms (\tau{n,m}(a_m)) for (m>n). If each (x^{(n)}) is (B_n)-letter‑balanced for some sequence ((B_n)), then (x) is factor‑balanced; if ((B_n)) is bounded, then (x) is uniformly factor‑balanced (Theorem 1.2). The proof reduces factor‑balancedness to letter‑balancedness after an appropriate desubstitution, using the notion of decisiveness to control the interaction between the S‑adic expansion and occurrences of factors.

Applying this result to primitive substitutive words (which are automatically linearly recurrent) yields a clean equivalence: a primitive substitution fixed point is uniformly factor‑balanced if and only if all its length‑2 factors are balanced (Proposition 1.3). This recovers and extends Adamczewski’s earlier theorem, showing that checking only length‑2 factors suffices for a large class of substitution systems.

The paper then turns to classical low‑complexity families. For Sturmian words (binary) and ternary Arnoux–Rauzy words, the authors prove that uniform factor‑balancedness holds precisely when the associated weak partial quotients are bounded (Theorem 1.4). This mirrors known characterizations of letter‑balancedness but now applies to the stronger factor‑balanced notion.

In the second major part, the relationship between factor‑balancedness and factor complexity is explored. Proposition 7.3 shows that any substitutive factor‑balanced word has linear factor complexity, and a more detailed structural description is given for such words. To demonstrate that the converse fails dramatically, the authors construct a Toeplitz word that is factor‑balanced yet exhibits exponential factor complexity (Theorem 1.5). Consequently, the symbolic dynamical system generated by this word has positive topological entropy and cannot have pure discrete spectrum. This provides a concrete counterexample to the conjecture raised by Arnoux, Berthé, Minervino, Steiner, and Thuswaldner (2025) that factor‑balancedness might imply a discrete spectrum.

The paper concludes with several open problems: (1) necessary conditions for linearly recurrent words to be (uniformly) factor‑balanced; (2) a full classification of uniformly factor‑balanced Arnoux–Rauzy words over arbitrary alphabet size; (3) whether every exponential growth rate of factor complexity can be realized by a factor‑balanced word; and (4) the existence of uniformly factor‑balanced words with super‑linear (but sub‑exponential) complexity.

Overall, the work advances the theory of factor‑balancedness by providing a versatile S‑adic criterion, linking it tightly to linear recurrence, delivering precise characterizations for important low‑complexity families, and clarifying its limitations through complexity‑based counterexamples. The results open new avenues for exploring the interplay between combinatorial regularity, dynamical spectra, and entropy in symbolic dynamics.