Let M be a positive integer and $q\in (1,M+1]$. A q-expansion of a real number x is a sequence $(c_i)=c_1{c}_2\cdots$ with $c_i\in \{0,1,\dots ,M\}$ such that $x={\sum }_{i=1}^{\infty }{c}_i{q}^{-i}$. Let ${\mathcal{U}}_q^j$ denote the set of numbers having exactly j expansions. Contrary to the unique expansions, we prove for each $j\ge 2$ that the set ${\mathcal{U}}_q^j$ is closed only if it is empty. Then we generalize an important example of Sidorov by exhibiting many non-trivial bases q for which the Hausdorff dimension of ${\mathcal{U}}_q^j$ is independent of j.

中文翻译：

---

多重展开的豪斯多夫维数

令M为正整数且$q\in (1,米+1]$. 甲q的实数的-expansion X是序列$(C_{\mathrm{一世}})=C_1{C}_2\cdots$ 和 $C_{\mathrm{一世}}\in \{0,1,\dots ,米\}$ 以至于 $X={\sum }_{\mathrm{一世}=1}^{\infty }{C}_{\mathrm{一世}}{q}^{-\mathrm{一世}}$. 让${\mathcal{你}}_q^j$表示恰好具有j 次扩展的一组数字。与唯一的展开相反，我们证明了每个$j\ge 2$ 那套 ${\mathcal{你}}_q^j$只有在它为空时才关闭。然后我们通过展示许多非平凡的基q来概括 Sidorov 的一个重要例子，其中的 Hausdorff 维数为${\mathcal{你}}_q^j$与j无关。