准晶与概周期性
Quasicrystals and almost periodicity
摘要：
本文在欧氏空间的一致离散子集空间$U$上引入拓扑${\cal T}$。假设$U$中的$S$允许唯一的自相关度量。$S$的衍射测度是纯原子的当且仅当$S$在$(U,{\cal T})$中是概周期的。这个结果将理想化的准晶与几乎周期性联系起来。在遍历点过程的上下文中，自相关测度是已知存在的。然后，衍射测度是纯原子的当且仅当动力系统具有纯点谱。作为一个例子，我们研究变形模型集。
英文摘要：
We introduce a topology ${\cal T}$ on the space $U$ of uniformly discrete subsets of the Euclidean space. Assume that $S$ in $U$ admits a unique autocorrelation measure. The diffraction measure of $S$ is purely atomic if and only if $S$ is almost periodic in $(U,{\cal T})$. This result relates idealized quasicrystals to almost periodicity. In the context of ergodic point processes, the autocorrelation measure is known to exist. Then, the diffraction measure is purely atomic if and only if the dynamical system has a pure point spectrum. As an illustration, we study deformed model sets.
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