Full dimensional sets of reals whose sums of partial quotients increase in certain speed

2016/10/10 by Ma, Liangang
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1610.02754

Abstract

For a real x∈(0,1)∖ℚ, let x=[a₁(x),a₂(x),⋯] be its continued fraction expansion. Let s_n(x)=∑_j=1ⁿ a_j(x). The Hausdorff dimensions of the level sets E_φ(n),α:=\x∈(0,1): lim_n→∞(s_n(x))/(φ(n))=α\ for α≥ 0 and a non-decreasing sequence \φ(n)\_n=1^∞ 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 φ(n) under which one can expect a 1-dimensional set E_φ(n),α. We give certain upper and lower bounds on the increasing speed of φ(n) when E_φ(n),α is of Hausdorff dimension 1 and a new class of sequences \φ(n)\_n=1^∞ such that E_φ(n),α is of full dimension. There is also a discussion of the problem in the irregular case.

Related

• Dimensions of certain sets of continued fractions with non-decreasing partial quotients
• A remark on Liao and Rams' result on distribution of the leading partial quotient with growing speed e^n^1/2 in continued fractions
• Hausdorff dimension of sets with restricted, slowly growing partial quotients
• A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions
• Metrical properties of exponentially growing partial quotients