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

Jörg Neunhäuserer

Published 2022-01-24Version 1

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 and set $B=\{x|\limsup_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension one. This result is strongly opposed to a result on the continued fraction expansion, here both sets have Hausdorff dimension $1/2$, see \cite{[GO],[LU]}. In addition we will find a dimension spectrum with respect to the base two expansion in the set $B$.