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 Rn. The importance of such quantitative statements is explicitly discussed in Jafar's monograph [12, §4.7.1].
Highlights
The present paper is motivated by a recent series of publications, including [11,12, 14,15,16,17,21,22,23], which utilise the theory of metric Diophantine approximation to develop new approaches in interference alignment, a concept within the field of wireless communication networks
The results obtained represent quantitative refinements of the Khintchine–Groshev Theorem that are relevant to the applications mentioned above
While the main content of the paper is purely number theoretic, in Appendix A we attempt to illustrate at a basic level the manner in which Diophantine Approximation plays a natural role in Interference Alignment
Summary
The present paper is motivated by a recent series of publications, including [11,12, 14,15,16,17,21,22,23], which utilise the theory of metric Diophantine approximation to develop new approaches in interference alignment, a concept within the field of wireless communication networks. In order to recall Khintchine’s theorem we first define the set W(ψ) of ψ-well approximable numbers. The applications typically require that κ(X) is independent of X This is impossible to guarantee with probability 1, that is on a set of full Lebesgue measure. To demonstrate this claim, let us define the following set: Bm,n(Ψ, κ) := X ∈ Mm,n : Xa > κΨ(a) ∀ a ∈ Zn \ {0}. Prior to giving a proof of this theorem recall that a measure μ on Mm,n is absolutely continuous with respect to Lebesgue measure if there exists a Lebesgue integrable function f : Mm,n → R+ such that for every Lebesgue measurable subset A of Mm,n, one has that μ(A) = f,. The results are obtained by exploiting the techniques of Bernik, Kleinbock and Margulis [8] originating from the seminal work of Kleinbock and Margulis [13] on the Baker–Sprindžuk conjecture
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