Number Theory Meets Wireless Communications: An Introduction for Dummies Like Us

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Title of host publicationNumber Theory Meets Wireless Communications
DateE-pub ahead of print - 22 Sep 2020
DatePublished (current) - 26 Jan 2021
Pages1-67
PublisherSpringer, Cham
EditorsVictor Beresnevich, Alister Burr, Bobak Nazer, Sanju Velani
Original languageEnglish
ISBN (Electronic)978-3-030-61303-7
ISBN (Print)978-3-030-61302-0

Publication series

NameMathematical Engineering

Abstract

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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