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

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          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.

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          Author and article information

          Journal
          29 June 2020
          Article
          2006.16358
          c986ab20-066d-4e87-9466-3ebffbd8ebcf

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          11J83, 11K60, 11J25, 37A17, 94A40, 94A15
          59 pages
          math.NT cs.IT math.DS math.IT

          Numerical methods,Differential equations & Dynamical systems,Information systems & theory,Number theory

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