Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896899 | Journal of Number Theory | 2018 | 23 Pages |
Abstract
For a real xâ(0,1)âQ, let x=[a1(x),a2(x),â¯] be its continued fraction expansion. Let sn(x)=âj=1naj(x). The Hausdorff dimensions of the level setsEÏ(n),α:={xâ(0,1):limnâââ¡sn(x)Ï(n)=α} for αâ¥0 and a non-decreasing sequence {Ï(n)}n=1â have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams et al. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on Ï(n) under which one can expect a 1-dimensional set EÏ(n),α. We give certain upper and lower bounds on the increasing speed of Ï(n) when EÏ(n),α is of Hausdorff dimension 1 and a new class of sequences {Ï(n)}n=1â such that EÏ(n),α is of full dimension. For an irregular sequence {Ï(n)}n=1â, a full dimensional set EÏ(n),α is impossible.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Liangang Ma,