𝛽-expansions with deleted digits for Pisot numbers 𝛽

An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ ( β , D ) Λ =Λ (β ,D) of real numbers with representations x = ∑ n = 1 ∞ d n β − n x=∑ ₙ₌₁^∞ d_n β ⁻ⁿ , where each d n ∈ D d_n ∈ D , a finite set of “digits”, and β > 0 β >0 is a Pisot number. The Hausdorff dimension is shown to be log ⁡ λ / log ⁡ β log λ /log β , where λ λ 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 β ,D is given.