2015/03/02 by Vilmos Komornik, Komornik, Vilmos, Derong Kong +3 · 1 citation
Computer Science · Mathematics · #11A63 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1503.00475
openalex publication_date 2015/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We fix a positive integer M, and we consider expansions in arbitrary real bases q>1 over the alphabet \0,1,...,M\. We denote by Uq the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of Uq for each q∈ (1,∞). Furthermore, we prove that the dimension function D:(1,∞)→[0,1] is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in (q',∞), where q' denotes the Komornik--Loreti constant: although D(q)>D(q') for all q>q', we have D'<0 a.e. in (q',∞). During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in β-expansions, and we determine for all M the Lebesgue measure and the Hausdorff dimension of the set of bases in which x=1 has a unique expansion.