Diophantine Approximation and applications in Interference Alignment

TL;DR

This paper extends the Khintchine-Groshev Theorem to non-degenerate submanifolds in Euclidean space, providing explicit quantitative results crucial for applications in interference alignment in electronics.

Contribution

It introduces and proves quantitative generalisations of the Khintchine-Groshev Theorem for submanifolds, advancing the mathematical foundation for interference alignment applications.

Findings

Quantitative generalisations of Khintchine-Groshev Theorem for submanifolds

Explicit bounds relevant for interference alignment

Enhanced understanding of Diophantine approximation in signal processing

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.