具有Borel窗口的弱模型集之谱
Spectrum of weak model sets with Borel windows
AI总结：
本文将弱模型集扩展壳同构于环面因子群的结果从紧窗口推广到任意可测相对紧Borel窗口，并讨论了衍射含义及一类对Mirsky测度通用的新例子。
摘要：
考虑一个弱模型集的扩展壳及其自然平移作用。赋予该扩展壳以Mirsky测度，这是一种特定的自然模式频率测度。已知扩展壳是某个群旋转的测度论因子，该群旋转被称为底环面。在文章《弱模型集的周期与因子》中，除其他结果外，我们证明了扩展壳同构于环面的一个因子群，其中弱模型集窗口的某些周期已被分解出去。这一结果当时是针对具有紧窗口的弱模型集证明的。在本注记中，我们论证同样的结果对任意可测且相对紧的窗口也成立。我们的论证关键依赖于Moody关于模型集中均匀分布的工作。我们还讨论了此类弱模型集的衍射所蕴含的结论，并讨论了一类对Mirsky测度而言为通用的新例子。
英文摘要：
Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article "Periods and factors of weak model sets" we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody's work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
