On the maximal length of consecutive zero digits of β-expansions

Let β > 1 be a real number. For any x ∈ [0, 1], let rn(x,β) be the maximal length of consecutive zero digits between the first n digits of x’s β-expansion. We prove that for Lebesgue almost all x ∈ [0, 1], limn→∞rn(x,β)/logβn=1. Also the Hausdorff dimensions of the related exceptional sets are determined.