Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772537 | Journal of Number Theory | 2017 | 15 Pages |
Abstract
For any real number β>1, the run-length function rn(β) is defined as the maximal length of consecutive zero digits amongst the first n digits in the β-expansion of 1. It was known that rn(β) grows to infinity with the speed logβâ¡n for Lebesgue almost all βâ(1,2). In this note, we quantify the size of the set of β for which rn(β) grows to infinity in a general speed. More precisely, for any strictly increasing function Ï:NâR+ with Ï(n) tending to +â and Ï(n)/n decreasing to 0 as nââ, we prove that for any real numbers 0â¤aâ¤bâ¤+â, the setEa,b={βâ(1,2):liminfnâârn(β)Ï(n)=a,limsupnâârn(β)Ï(n)=b} is of full Hausdorff dimension.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Chunyun Cao, Yuanhong Chen,