On the dimension of certain sets araising in the base two expansion

TL;DR

This paper investigates the Hausdorff dimension of sets defined by the growth of digits in a specific base-two expansion, revealing that the set where digits tend to infinity has dimension zero, contrasting with continued fraction cases.

Contribution

It establishes the Hausdorff dimension of the set where digits tend to infinity in a base-two expansion and constructs subsets with full dimension within a limsup set, also analyzing the dimension spectrum.

Findings

Set of points with digits tending to infinity has Hausdorff dimension zero.

Constructed subsets of points with unbounded digits have Hausdorff dimension one.

Identified a dimension spectrum within the limsup set of unbounded digits.

Abstract

We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is opposed to a result on the continued fraction expansion, here $A$ has Hausdorff dimension $1/2$, see \cite{[GO]}. Furthermore we construct subsets of $B=\{x|\limsup_{i\to\infty}d_{i}(x)=\infty\}$ which have Hausdorff dimension one and find a dimension spectrum in set $B$.