Let β > 1 and x ∈ [0, 1) be two real numbers. For any y ∈ [0, 1), the maximal run-length function r x (y, n) (with respect to x) is defined to be the maximal length of the prefix of x's β-expansion which appears in the first n digits of y's. In this paper, we study the metric properties of the maximal run-length function and apply them to the hitting time, which generalises many known results. In the meantime, the fractal dimensions of the related exceptional sets are also determined.

中文翻译：

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β-动力系统中实数的最大游程函数

设 β > 1 且 x ∈ [0, 1) 是两个实数。对于任何 y ∈ [0, 1)，最大游程长度函数 rx (y, n)（相对于 x）被定义为 x 的 β-展开的前缀的最大长度，该前缀出现在前 n 位你的。在本文中，我们研究了最大游程长度函数的度量属性并将它们应用于击球时间，这概括了许多已知结果。同时，还确定了相关异常集的分形维数。