Riesz基、Meyer准晶体和有界余集
Riesz bases, Meyer's quasicrystals, and bounded remainder sets
摘要：
我们考虑$\mathbb{R}^d$中频率属于简单准晶的指数系统。我们问，如果存在域$S$在$\mathbb{R}^d$，其中允许这样的系统作为空间$L^2(S)$的Riesz基。我们证明，答案取决于一个算术条件的准晶。证明是基于连接的问题，多维无理旋转的差异，特别是有界剩余集的理论。特别是它表明，任何有界剩余集承认Riesz基础的指数。这扩展到几个维度（和非周期性设置）的结果在维度1较早获得。
英文摘要：
We consider systems of exponentials with frequencies belonging to simple quasicrystals in $\mathbb{R}^d$. We ask if there exist domains $S$ in $\mathbb{R}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several dimensions (and to the non-periodic setting) the results obtained earlier in dimension one.
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。
