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NUMBER THEORY PROBLEMS RELATED TO THE SPECTRUM OF CANTOR-TYPE MEASURES WITH CONSECUTIVE DIGITS
Published online by Cambridge University Press: 10 June 2020
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
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(pb,S)=\bigg\{\mathop{\sum }_{j=0}^{\text{finite}}(pb)^{j}s_{j}:s_{j}\in S\bigg\},\end{eqnarray}$$
$S=pD$. 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.
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 113 - 123
- Copyright
- © 2020 Australian Mathematical Publishing Association Inc.
Footnotes
The research is supported in part by the NNSF of China (No. 11831007) and by Hunan Provincial Innovation Foundation for Postgraduates (CX20190322).
References
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