2015/08/25 by Karma Dajani, Kan Jiang, Dajani, Karma +5 · 1 citation
Computer Science · Mathematics · #11K55 #37B10 #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary: 11A63 #Secondary: 10K50 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1508.06138
openalex publication_date 2015/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For q>1 we consider expansions in base q over the alphabet \0,1,q\. Let Uq be the set of x which have a unique q-expansions. For k=2, 3,⋯,ℵ0 let Bk be the set of bases q for which there exists x having k different q-expansions, and for q∈ Bk let Uq(k) be the set of all such x's which have k different q-expansions. In this paper we show that Bℵ0=[2,∞), Bk=(qc,∞) \textrmfor any k≥ 2, where qc≈ 2.32472 is the appropriate root of x3-3x2+2x-1=0. Moreover, we show that for any positive integer k≥ 2 and any q\inBk the Hausdorff dimensions of Uq(k) and Uq are the same, i.e., dimHUq(k)=dimHUq \textrmfor any k≥ 2. Finally, we conclude that the set of x having a continuum of q-expansions has full Hausdorff dimension.