A result on the maximal length of consecutive 0 digits in $beta$- expansionsexpansions

Abstract

Let β > 1 be a real number. For any x ∈ [0, 1] , let rn(x, β) be the maximal length of consecutive zero digits in the first n digits of the β -expansion of x . In this note, it is proved that for any 0 < a < b < +∞ , the set Ea,b = {x ∈ [0, 1] : lim inf n→∞ rn(x, β) logβ n = a, lim sup n→∞ rn(x, β) logβ n = b} has the full Hausdorff dimension.