2016/06/13 by Derong Kong, Kong, Derong, Wenxia Li +5 · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #math.DS #math.NT #msc:11A63 #msc:28A78 #msc:37B10 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1606.03791
16 pages. To appear in Monatshefte fur Mathematik (2017)
arxiv created 2017/04/02 · arxiv updated 2017/04/04
Given a positive integer M and a real number q >1, a q-expansion of a real number x is a sequence (ci)=c1c2⋯ with (ci) ∈ \0,…,M\^∞ such that x=∑i=1∞ ciq-i. It is well known that if q ∈ (1,M+1], then each x ∈ Iq:=[0,M/(q-1)] has a q-expansion. Let U=U(M) be the set of univoque bases q>1 for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of U and to show that the Hausdorff dimension of the set of numbers x ∈ Iq with a unique q-expansion changes the most if q "crosses" a univoque base. Denote by B2=B2(M) the set of q ∈ (1,M+1] such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (2009) and prove that dimH(B2∩(q',q'+δ))>0 \textrmfor any δ>0, where q'=q'(M) is the Komornik-Loreti constant.