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

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 $B$.