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

TL;DR

This paper investigates the conditions on the growth rate of partial quotient sums in continued fractions that lead to sets of full Hausdorff dimension, extending previous work on level sets of these sums.

Contribution

It provides bounds and new classes of sequences for which the level sets of partial quotient sums have Hausdorff dimension one.

Findings

Identifies growth conditions on unction or full Hausdorff dimension

Establishes upper and lower bounds on or dimension 1

Proposes new sequence classes with full dimension sets

Abstract

For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Let $s_n(x)=\sum_{j=1}^n a_j(x)$. The Hausdorff dimensions of the level sets $E_{\varphi(n),\alpha}:=\{x\in(0,1): \lim_{n\rightarrow\infty}\frac{s_n(x)}{\varphi(n)}=\alpha\}$ for $\alpha\geq 0$ and a non-decreasing sequence $\{\varphi(n)\}_{n=1}^\infty$ have been studied by E. Cesaratto, B. Vall\'ee, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams \emph{et al}. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on $\varphi(n)$ under which one can expect a $1$-dimensional set $E_{\varphi(n),\alpha}$. We give certain upper and lower bounds on the increasing speed of $\varphi(n)$ when $E_{\varphi(n),\alpha}$ is of Hausdorff dimension 1 and a new class of sequences $\{\varphi(n)\}_{n=1}^\infty$ such that $E_{\varphi(n),\alpha}$ is of full dimension. There is also a discussion of the problem in the irregular case.