Number Theory Problems Related to the Spectrum of Cantor-Type Measures With Consecutive Digits

Abstract

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.