Maximal run-length function for real numbers in beta-dynamical system

Abstract

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 de ned to be the maximal length of the pre x of x's β-expansion which appears in the rst 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.