The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by Čech cohomology of the tiling space) and the spectral properties (of Hamiltonians defined on such tilings) defined by $K$-theory, and to show their equivalence in dimensions $\le$ 3. A theorem makes precise the conditions for this relationship to hold. It can be viewed as an extension of the “Bloch Theorem” to a large class of aperiodic tilings. The idea underlying this result is based on the relationship between cohomology and $K$-theory traces and their equivalence in low dimensions.

中文翻译：

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关于非周期性平铺的衍射和光谱数据：走向布洛赫定理

本文的目的是显示平铺的结构（衍射图样）方面（由平铺空间的Čech谐函数描述）和由平铺定义的哈密顿量（由哈密顿量定义）之间的所有维度之间的关系。 $ķ$-理论，并在尺寸上证明它们的等效性 $\le$3.一个定理精确地说明了保持这种关系的条件。可以将其视为“布洛赫定理”（Bloch Theorem）的扩展，以扩展为 各种非周期性平铺图。该结果的基本思想是基于同调与$ķ$-理论轨迹及其在低维中的等效性。