Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

中文翻译：

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相对分岔集和单义基的局部维数

修复一个字母$A=\{0,1,\ldots ,M\}$和$M\in \mathbb{N}$. 独特的集合$\mathscr{U}$碱基数$q\in (1,M+1)$其中数$1$在字母表上有一个独特的扩展$澳元已经很好地研究了。它的 Lebesgue 量度为零，但 Hausdorff 量度为一。本文描述了集合中的点如何$\mathscr{U}$分布在区间上$(1,M+1)$通过确定极限$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q- \unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$对所有人$q\in (1,M+1)$. 我们特别表明$f(q)>0$当且仅当$q\in \overline{\mathscr{U}}\反斜杠 \mathscr{C}$， 在哪里$\mathscr{C}$是 Hausdorff 维数为零的不可数集，并且$f$在它消失的那些（并且只有那些）点上是连续的。此外，我们引入了可数族的成对不相交子集$\mathscr{U}$称为相对分岔集，并用它们给出交集的豪斯多夫维数的显式表达式$\mathscr{U}$以任意间隔，回答 Kalle 的问题等人[关于独特扩展的分岔集。阿里斯学报。188(2019), 367–399]。最后，利用本文开发的方法对第一作者提出的一个问题[On univoque and strong univoque sets。进阶。数学。308(2017), 575–598] 在强单义集上。