A mass transference principle for systems of linear forms and its applications

Abstract

In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine-Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira [8].

A general mass transference principle

“…Let (X, d), F, Υ, g and f be as given in Theorem 1 and assume that the hypotheses of Theorem 1 hold. Then, for any ball B in X and any G ∈ N, there exists a finite collection Similarly to [1,Lemma 4], the collection K G,B here is a collection of balls drawn from the families Φ j (B). These balls correspond to the lim sup set Λ(Υ).…”

“…As mentioned previously, from each of the balls in K G,B we wish to extract a collection of balls corresponding to Λ(Υ). The desired properties and existence of such collections are summarised in the following lemma, which constitutes the required analogue of [1,Lemma 5] in this setting. Lemma 6.…”

“…The construction of the Cantor set we present here is an adaptation of that given in [1] and [5]. For ease of comparison we will generally adopt the notation used in [1].…”

Metrical theorems on systems of affine forms

“…The most fundamental result in metric Diophantine approximation is the Khintchine-Groshev theorem which gives an elegant answer to the question of the 'size' of the set in terms of Lebesgue or Hausdorff measure. For the modernised version of the Khintchine-Groshev theorem for W 0 (m, n; ψ) we refer to [3] and for W α (m, n; ψ) we refer to [1]. The doubly metric version of the Khintchine-Groshev theorem not only requires weaker assumptions than the homogeneous analogue but is also considerably easier to prove, see [38,Theorem 15], or [10, Theorem VII.II] for the special case of simultaneous approximation.…”

“…Our new results on the metric properties of the sets W (m, n, u; ψ) and the fibers W α (m, n, u; ψ) = {X ∈ R mn : (X, α) ∈ W (m, n, u; ψ)} are proved using the methods developed in [27], including the appeal to the important tools of Beresnevich and Velani, the mass transference principle [4] and its generalization to linear forms [1,5]. The adaptation of the methods to the present setup requires some work, so rather than stating just the differences with the manuscript [27], we have chosen to present the present work in a more self-contained manner.…”