Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5774519 | Journal of Mathematical Analysis and Applications | 2017 | 10 Pages |
Abstract
Let β>1 be a real number. For any xâ[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}nâ¥1 be a non-decreasing sequence of integers and defineE({δn}nâ¥1)={xâ[0,1]:limsupnâârn(x,β)δn=1}. In this paper, we show thatdimHâ¡E({δn}nâ¥1)=maxâ¡{0,1âliminfnââδn⧸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 liminfnââδnn=0, then the setEmax({δn}nâ¥1)={xâ[0,1]:liminfnâârn(x,β)δn=0,limsupnâârn(x,β)δn=+â} has full Hausdorff dimension.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Jia Liu, Meiying Lü,