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.

翻译：在无限词的研究中，各种平衡性概念为字母或因子出现的规律性提供了定量度量，并在数学与理论计算机科学的多个领域中得到应用。本文研究了因子平衡性与一致因子平衡性，并作出两项主要贡献。首先，我们建立了无限词具有（一致）因子平衡性的一般充分条件，特别适用于任意给定的线性递推词。这些条件通过$\mathcal{S}$-adic表示进行表述，并推广了Adamczewski关于本原替换词的结果——该结果表明长度为2的因子平衡性已蕴含一致因子平衡性。作为我们判据的应用，我们刻画了一致因子平衡的Sturmian词与三元Arnoux--Rauzy词，精确对应于具有有界弱部分商的词。我们的第二项主要贡献是研究了因子平衡性与因子复杂度之间的关系。特别地，我们分析了非本原替换情形，并构造了一个具有指数因子复杂度的因子平衡词示例，从而对Arnoux、Berthé、Minervino、Steiner和Thuswaldner于2025年提出的关于平衡性与离散谱关系的问题取得了进展。