The Hausdorff dimension of $\lambda$-expansions with deleted digits

Abstract:

In this article we examine the continuity of the Hausdorff dimension of the one parameter family of Cantor sets $\Lambda (\lambda ) = \{ \sum \nolimits _{k = 1}^\infty {{i_k}{\lambda ^k}:{i_k} \in S\} }$, where $S \subset \{ 0,1, \ldots ,(n - 1)\}$. In particular, we show that for almost all (Lebesgue) $\lambda \in [\tfrac {1} {n},\tfrac {1} {l}]$ we have that ${\dim _H}(\Lambda (\lambda )) = \frac {{\log l}} {{ - \log \lambda }}$ where $l = \operatorname {Card} (S)$. In contrast, we show that under appropriate conditions on $S$ we have that for a dense set of $\lambda \in [\tfrac {1} {n},\tfrac {1} {l}]$ we have ${\dim _H}(\Lambda (\lambda )) < \frac {{\log l}} {{ - \log \lambda }}$.