  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.
