We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

中文翻译：

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一级马勒函数：渐近、应用和推测

我们将一阶马勒函数的径向渐近性完整描述为$z$接近度数单位的根$k^{n}$， 在哪里$k$是马勒函数的基，以及一些关于超越和代数独立的应用。例如，我们证明了 Thue-Morse 序列的生成函数和具有非零马勒特征值的任何马勒函数（到相同的基）在代数上独立于$\mathbb{C}(z)$. 最后，我们讨论了单位圆上通用点的渐近界。