A note on metric inhomogeneous Diophantine approximation

Abstract

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

Metric inhomogeneous Diophantine approximation in positive characteristic

“…This result also extends results of Dickinson [5] and Dodson [6] to the setting of formal Laurent series. Corollary 1.8.…”

“…We will also deduce Hausdorff dimension versions of some of these theorems. In particular, we extend results of Dickinson [5], Dodson [6] and Levesley [12] to the present setting. As a corollary of our results, we extend a recent result of Ma and Su [13] to higher dimensions.…”

A Note on Inhomogeneous Diophantine Approximation

“…REMARK 2. Dodson [7] and Levesley [8] have determined the Hausdorff dimension of two sets which are closely related to V v (α). Namely, for any v ≥ 1, they have, respectively, established that dim{(α, ξ ) ∈ R 2 : nα − ξ < n −v holds for infinitely many n ∈ N} = 1 + 2 v + 1 and, for any given real number ξ , that dim{α ∈ R : nα − ξ < n −v holds for infinitely many n ∈ N} = 2 v + 1 , which can be viewed as a 'dual' form of Theorem 1.…”

On simultaneous inhomogeneous Diophantine approximation

“…. , ξ k ), and for any real numbers v > k and w > 1/k, the Hausdorff dimensions of the sets To complement this result, we mention that, in the "doubly metric" case, Dodson [7] established that, for real numbers v > k and w > 1/k, the Hausdorff dimensions of the sets…”

“…In the "doubly metric" case and in the first "singly metric" case mentioned above, satisfactory answers have been given by Dodson [7] and Levesley [13], respectively. However, no multidimensional extension of the statements established in [14] and [4] has been studied up to now, and it is the purpose of the present work to report various results on this question.…”