Run-length function of the beta-expansion of a fixed real number

Abstract

For any real numbers x∈[0,1] and β>1, let rn(x,β) be the maximal length of consecutive 0's in the first n digits of the beta-expansion of x in base β. The run-length function rn(x,β) has been well studied for a fixed base β>1 or a fixed real number x=1. In this paper, we prove that for any x∈(0,1), the setDx={β>1:limn→∞⁡rn(x,β)logβ⁡n=1} is of full Lebesgue measure in (1,+∞). When the exceptional set is considered, we prove that for any real numbers 0≤a≤b≤+∞, the setDx(a,b)={β>1:lim infn→∞rn(x,β)logβ⁡n=a,lim supn→∞rn(x,β)logβ⁡n=b} is of full Hausdorff dimension. We also determine the Hausdorff dimension of the setFx(c,d)={β>1:lim infn→∞rn(x,β)n=c,lim supn→∞rn(x,β)n=d} for any real numbers 0≤c≤d≤1.