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

We show that for the base two expansion $$x=\sum _{i=1}^{\mathrm{\infty }}{2}^{-(d_1(x)+d_2(x)+\cdots +d_i(x))}$$ with $x\in (0,1]$ and $d_i(x)\in \mathbb{N}$ the set $A=\{x\mathrm{\mid }{\lim }_{i\to \mathrm{\infty }}{d}_i(x)=\mathrm{\infty }\}$ has Hausdorff dimension zero, this is opposed to a result on the continued fraction expansion, here $A$ has Hausdorff dimension $1\mathrm{/}2$, see \cite{[GO]}. Furthermore we construct subsets of $B=\{x\mathrm{\mid }{\mathrm{lim\hspace{0.17em}sup}}_{i\to \mathrm{\infty }}{d}_i(x)=\mathrm{\infty }\}$ which have Hausdorff dimension one and find a dimension spectrum in set $B$.