[en] 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 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.
