Univoque bases and Hausdorff dimension

Abstract

Given a positive integer $M$ and a real number $q>1$, a \emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1{c}_2\cdots$ with $(c_i)\in \{0,\dots ,M{\}}^{\infty }$ such that \[x=\sum_{i=1}^{\infty} c_iq^{-i}.\] It is well known that if $q\in (1,M+1]$, then each $x\in I_q:=\left[0,M/(q-1)\right]$ has a $q$-expansion. Let $\mathcal{U}=\mathcal{U}(M)$ be the set of \emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion. The main object of this paper is to provide new characterizations of $\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $x\in I_q$ with a unique $q$-expansion changes the most if $q$ "crosses" a univoque base. Denote by ${\mathcal{B}}_2={\mathcal{B}}_2(M)$ the set of $q\in (1,M+1]$ 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 \[\dim_H(\mathcal{B}_2\cap(q',q'+δ))>0\quad\textrm{for any}\quad δ>0,\] where $q^{\prime }=q^{\prime }(M)$ is the Komornik-Loreti constant.

16 pages. To appear in Monatshefte fur Mathematik (2017)

References in corpus (3)

• Expansions in non-integer bases: lower, middle and top orders
• Greedy and quasi-greedy expansions in non-integer bases
• On univoque Pisot numbers

Cited by in corpus (3)

• Entropy, topological transitivity, and dimensional properties of unique $q$-expansions
• On the bifurcation set of unique expansions
• Algebraic sums and products of univoque bases