The Metric Theory of Mixed Type Linear Forms

Abstract

In this paper the metric theory of Diophantine approximation of linear forms that are of mixed type is investigated. Khintchine–Groshev theorems are established together with the Hausdorff measure generalizations. The latter includes the original dimension results obtained in [H. Dickinson, The Hausdorff dimension of sets arising in metric Diophantine approximation, Acta Arith.68(2) (1994) 133–140] as special cases.

A problem in non-linear Diophantine approximation

“…In order to prove Proposition 2 we wish to estimate the measure of the intersections ℓ a,b ∩ B for (a, b) ∈ N n,m . We first verify condition (12). To do this, for each (a, b) ∈ N n,m we first require to estimate the number of integers c for which ℓ a,b (c) ∩ B = ∅.…”

A note on the weighted Khintchine-Groshev Theorem

“…For the sake of completeness we mention that a Hausdorff measure version of Theorem 1 can be straightforwardly established using the Mass Transference Principle of [4] along with the 'slicing' technique [5]. The slicing technique is broad ranging and has been successfully employed in various related settings for a similar purpose [8,13,14].…”

Metrical theorems on systems of affine forms

“…is well studied by various authors in [14,16,24,27,29]. The set arose in the literature in part due to its connections with Kolmogorov-Arnold-Moser theory [2,19], linearization of germs of complex analytic diffeomorphisms of C m near a fixed point [2,20], operator theory [12,13] and recently discovered applications in signal processing [33,35].…”

“…Remark. This Lemma is an analogue of [14, Lemma 1] but we should point out that the calculation in [14] has a small error: if we consider the set of matrices X such that the m − 1 certain prescribed columns of X are equal to certain prescribed values, then it is not true that the dimension of this set is always equal to the number claimed. It may be higher, e.g.…”