Diophantine Approximation and applications in Interference Alignment

Abstract: This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of $\mathbb{R}^n$. The importance of such quantitative statements is explicitly discussed in Section 4.7.1 of Jafar's monograph `Interference Alignment - A New Look at Signal Dimensions in a Communication Network', Foundations and Trends in Communications and Information Theory, Vol. 7, no. 1, 2010.