Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal U(x)$ such that when $x$ passes the first critical value the set $\mathcal U(x)$ changes from a set with positive Hausdorff dimension to a countable set, and when $x$ passes the second critical value the set $\mathcal U(x)$ changes from an infinite set to a singleton. Denote by $\mathbf U(x)$ the set of all unique $q$-expansions of $x$ for $q\in\mathcal U(x)$. We give the Hausdorff dimension of $\mathbf U(x)$ and show that the dimensional function $x\mapsto\dim_H\mathbf U(x)$ is a non-increasing Devil's staircase. Finally, we investigate the topological structure of $\mathcal U(x)$. In contrast with $x=1$ that $\mathcal U(1)$ has no isolated points, we prove that for typical $x>0$ the set $\mathcal U(x)$ contains isolated points.

中文翻译：

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实数的唯一基：局部维数、魔鬼阶梯和孤立点

用 $\mathbf U(x)$ 表示 $q\in\mathcal U(x)$ 的 $x$ 的所有唯一的 $q$-展开的集合。我们给出 $\mathbf U(x)$ 的 Hausdorff 维数，并证明维函数 $x\mapsto\dim_H\mathbf U(x)$ 是一个非递增的魔鬼阶梯。最后，我们研究了 $\mathcal U(x)$ 的拓扑结构。与 $x=1$ 相比，$\mathcal U(1)$ 没有孤立点，我们证明对于典型的 $x>0$ 集合 $\mathcal U(x)$ 包含孤立点。