Number Theory meets Wireless Communications: an introduction for dummies like us

TL;DR

This paper introduces Diophantine approximation theory to wireless communications, demonstrating how number theoretic concepts can analyze and improve the Degrees of Freedom in communication channels, with new results on channel coefficient conditions.

Contribution

It extends the understanding of Degrees of Freedom in wireless channels by showing they can be achieved for all channel coefficients except a small subset, using new number theoretic approaches.

Findings

Achieved total DoF for all channel coefficients except a small subset.

Introduced the concept of jointly non-singular points in the analysis.

Improved previous results on the DoF of a two-user X-channel.

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 $\delta$-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.