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 $B=\{x|{\mathrm{limsup}}_{i\to \infty }{d}_i(x)=\infty \}$ which have Hausdorff dimension one and find a dimension spectrum in set $B$.

arXiv commentThe results will be published in a more general setting in another paper