论文标题
数字理论符合无线通信：像我们这样的假人的介绍
Number Theory meets Wireless Communications: an introduction for dummies like us
论文作者
论文摘要
在本章中，我们通过与无线通信相关的信息理论中的一系列基本示例介绍了二芬太汀近似理论。特别是，我们讨论了Dirichlet的定理，差异很差的点，Dirichlet可改善和奇异点，包括Khintchine-groshev定理，包括Khintchine-groshev定理以及二磷剂理论的二聚体近似值的度量（概率）理论。我们探讨用于分析沟通特征（例如自由度（DOF））的各种理论方法。特别是，我们改善了Motahari等人关于两用户X通道的DOF的结果。本质上，我们表明，除了与环境空间相比，严格尺寸小的子集以外，所有（几乎不是所有）的通道系数选择可以实现总DOF。该改进利用了我们引入的共同非单明点的概念，也是Kadyrov等人在晶格空间中质量$δ$ escape上的一般结果。我们还讨论了后续的开放问题，这些问题融合了张的突破，并且在奇异点的尺寸上更加普遍。
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.
