$\beta$-expansions with deleted digits for Pisot numbers $\beta$

Abstract:

An algorithm is given for computing the Hausdorff dimension of the set(s) $\Lambda =\Lambda (\beta ,D)$ of real numbers with representations $x=\sum _{n=1}^\infty d_n \beta ^{-n}$, where each $d_n \in D$, a finite set of “digits”, and $\beta >0$ is a Pisot number. The Hausdorff dimension is shown to be $\log \lambda /\log \beta$, where $\lambda$ is the top eigenvalue of a finite 0-1 matrix $A$, and a simple algorithm for generating $A$ from the data $\beta ,D$ is given.