𝛽-expansions with deleted digits for Pisot numbers 𝛽

Source abstract

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