数论与无线通信：为我们这样的傻瓜介绍
Number Theory meets Wireless Communications: an introduction for dummies like us
摘要：
在这一章中，我们通过一系列与无线通信相关的信息论的基本例子来介绍丢番图逼近的理论。特别地，我们讨论了Dirichlet定理，差逼近点，Dirichlet可改进点和奇点，丢番图逼近的度量（概率）理论，包括Khintchine-Groshev定理和流形上的丢番图逼近理论。我们探讨了各种数论方法用于分析通信特性，如自由度（DoF）。特别是，我们改进了Motahari等人关于两个用户X通道的DoF的结果。从本质上讲，我们表明，总自由度可以实现所有（而不是几乎所有）的信道系数的选择，除了一个子集的严格较小的尺寸比周围的空间。改进利用了我们引入的联合非奇点的概念和Kadyrov等人关于格空间中质量的$δ$逃逸的一般结果。我们还讨论了后续开放的问题，将突破张和更普遍的达斯等人的尺寸的一组奇点。
英文摘要：
In this chapter we introduce the theory of Diophantine approximation via a series of basic examples from information theory relevant to wireless communications. In particular, we discuss Dirichlet's theorem, badly approximable points, Dirichlet improvable and singular points, the metric (probabilistic) theory of Diophantine approximation including the Khintchine-Groshev theorem and the theory of Diophantine approximation on manifolds. We explore various number theoretic approaches used in the analysis of communication characteristics such as Degrees of Freedom (DoF). In particular, we improve the result of Motahari et al regarding the DoF of a two-user X-channel. In essence, we show that the total DoF can be achieved for all (rather than almost all) choices of channel coefficients with the exception of a subset of strictly smaller dimension than the ambient space. The improvement utilises the concept of jointly non-singular points that we introduce and a general result of Kadyrov et al on the $δ$-escape of mass in the space of lattices. We also discuss follow-up open problems that incorporate a breakthrough of Cheung and more generally Das et al on the dimension of the set of singular points.
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