For integers $p,b\geq 2$, let $D=\{0,1,\ldots ,b-1\}$ be a set of consecutive digits. It is known that the Cantor measure $\unicode[STIX]{x1D707}_{pb,D}$ generated by the iterated function system $\{(pb)^{-1}(x+d)\}_{x\in \mathbb{R},d\in D}$ is a spectral measure with spectrum $$\begin{eqnarray}\unicode[STIX]{x1D6EC}(pb,S)=\bigg\{\mathop{\sum }_{j=0}^{\text{finite}}(pb)^{j}s_{j}:s_{j}\in S\bigg\},\end{eqnarray}$$ where $S=pD$. We give conditions on $\unicode[STIX]{x1D70F}\in \mathbb{Z}$ under which the scaling set $\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6EC}(pb,S)$ is also a spectrum of $\unicode[STIX]{x1D707}_{pb,D}$. These investigations link number theory and spectral measures.

中文翻译：

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与具有连续数字的康托尔式测度谱有关的数论问题

对于整数$p,b\geq 2$， 让$D=\{0,1,\ldots ,b-1\}$是一组连续的数字。众所周知，康托尔测度$\unicode[STIX]{x1D707}_{pb,D}$由迭代函数系统生成$\{(pb)^{-1}(x+d)\}_{x\in \mathbb{R},d\in D}$是光谱测量$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(pb,S)=\bigg\{\mathop{\sum }_{j=0}^{\text{finite}}(pb)^{ j}s_{j}:s_{j}\in S\bigg\},\end{eqnarray}$$在哪里$S=pD$. 我们给出条件$\unicode[STIX]{x1D70F}\in \mathbb{Z}$在哪个尺度下设置$\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6EC}(pb,S)$也是一个谱$\unicode[STIX]{x1D707}_{pb,D}$. 这些研究将数论和谱测量联系起来。