Hausdorff dimension of some sets arising by the run-length function of β-expansions

Abstract

Let β>1 be a real number. For any x∈[0,1], the run-length function rn(x,β) is defined as the length of the longest run of 0's amongst the first n digits in the β-expansion of x. Let {δn}n≥1 be a non-decreasing sequence of integers and defineE({δn}n≥1)={x∈[0,1]:limsupn→∞rn(x,β)δn=1}. In this paper, we show thatdimH⁡E({δn}n≥1)=max⁡{0,1−liminfn→∞δn⧸n}. Using the same method, we also study a class of extremely refined subset of the exceptional set in Erdös–Rényi limit theorem. Precisely, we prove that if liminfn→∞δnn=0, then the setEmax({δn}n≥1)={x∈[0,1]:liminfn→∞rn(x,β)δn=0,limsupn→∞rn(x,β)δn=+∞} has full Hausdorff dimension.