[Paper Review] Univoque bases and Hausdorff dimension

Derong Kong, Wenxia Li+2 more|arXiv (Cornell University)|Jun 13, 2016

Mathematical Dynamics and Fractals18 references3 citations

TL;DR

This paper provides new characterizations of univoque bases and their relation to Hausdorff dimension in non-integer base expansions. It proves that the Hausdorff dimension of the set of numbers with unique q-expansions changes significantly when q crosses a univoque base, and resolves a question by Sidorov by showing that the set of bases with exactly two expansions has positive local Hausdorff dimension near the Komornik-Loreti constant.

ABSTRACT

Given a positive integer $M$ and a real number $q >1$, a \emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $(c_i) \in \{0,\ldots,M\}^\infty$ such that \[x=\sum_{i=1}^{\infty} c_iq^{-i}.\] It is well known that if $q \in (1,M+1]$, then each $x \in I_q:=\left[0,M/(q-1) ight]$ 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 \[\dim_H(\mathcal{B}_2\cap(q',q'+δ))>0\quad extrm{for any}\quad δ>0,\] where $q'=q'(M)$ is the Komornik-Loreti constant.

Motivation & Objective

• To characterize univoque bases q > 1 for which 1 has a unique q-expansion over the digit set {0, ..., M}.
• To investigate how the Hausdorff dimension of the set of numbers with unique q-expansions changes when q crosses a univoque base.
• To resolve a question by Sidorov (2009) regarding the local Hausdorff dimension of the set of bases with exactly two expansions.
• To establish equivalent conditions for q to belong to the closure of the univoque set using Hausdorff dimension of expansion differences and local intersections.
• To analyze the structure of univoque sets via symbolic dynamics and generalized Thue-Morse sequences, particularly focusing on irreducible subshifts of finite type.

Proposed method

• Introduces the univoque set $\mathcal{U}_q$ as the set of real numbers in $[0, M/(q-1)]$ with a unique q-expansion in base $q > 1$.
• Uses the natural projection $\pi_{M+1}$ to map sequences of digits to real numbers, enabling the study of Hausdorff dimension of sets of expansions.
• Applies symbolic dynamics and subshifts of finite type, particularly constructing irreducible subshifts $X_A^{(n)}$ over states $\{\xi_n, \xi_n^-, \overline{\xi_n}, \overline{\xi_n^-}\}$ to model univoque expansions.
• Employs generalized Thue-Morse sequences $\theta = (\theta_i)$ to construct sequences in $\Gamma_n'$ that correspond to univoque bases via recursive inequalities and lexicographic ordering.
• Uses the fact that $\dim_H \pi_{M+1}(X_A^{(n)}) = \frac{\log((1+\sqrt{5})/2)}{2^n p \log(M+1)} > 0$ to establish positive dimension for univoque sets near certain bases.
• Applies results from de Vries (2008) and Komornik et al. (2007) on the Devil’s staircase behavior of the dimension function $q \mapsto \dim_H \mathcal{U}_q$.

Experimental results

Research questions

• RQ1 What characterizations of univoque bases $q \in \mathcal{U}(M)$ can be given in terms of the Hausdorff dimension of the difference set $\mathcal{U}_r' \setminus \mathcal{U}_q'$ for $r > q$?
• RQ2 How does the Hausdorff dimension of the set of numbers with unique q-expansions behave when $q$ crosses a univoque base?
• RQ3 Does the set $\mathcal{B}_2$ of bases with exactly two distinct expansions have positive local Hausdorff dimension near the Komornik-Loreti constant $q'$?
• RQ4 What is the relationship between the closure of the univoque set $\overline{\mathcal{U}}$ and the local dimension of $\mathcal{U}_q'$ and $\mathcal{U}$ near $q$?
• RQ5 Can the structure of univoque expansions be analyzed using generalized Thue-Morse sequences and irreducible subshifts of finite type?

Key findings

• The Hausdorff dimension of $\pi_{M+1}(\mathcal{U}_r' \setminus \mathcal{U}_q')$ is positive for all $r > q$ if and only if $q$ is a univoque base, establishing a new characterization of $\mathcal{U}$.
• The Hausdorff dimension of $\mathcal{U} \cap (q, r)$ is positive for all $r > q$ if and only if $q$ is a univoque base, providing a local dimension-based characterization of $\mathcal{U}$.
• The set $\mathcal{B}_2 \cap (q', q' + \delta)$ has positive Hausdorff dimension for any $\delta > 0$, answering Sidorov’s (2009) question affirmatively.
• For $q \in \overline{\mathcal{U}} \setminus (\bigcup \{q_0^*\} \cup \{q'\})$, the Hausdorff dimension of $\pi_{M+1}(\mathcal{U}_q' \setminus \mathcal{U}_p')$ is positive for all $p < q$, providing a new characterization of such $q$.
• The Hausdorff dimension of $\mathcal{U} \cap (p, q)$ is positive for all $p < q$ if and only if $q$ is in $\overline{\mathcal{U}} \setminus (\bigcup \{q_0^*\} \cup \{q'\})$, offering a local dimension criterion for membership in the closure of the univoque set.
• The dimension of the projection of the subshift $X_A^{(n)}$ is $\frac{\log((1+\sqrt{5})/2)}{2^n p \log(M+1)} > 0$, which is used to prove the positivity of local dimension in univoque sets near certain bases.