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

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 $B$.