Metric Diophantine Approximation: Aspects of Recent Work

Abstract

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarník, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.

Inhomogeneous Dual Diophantine Approximation on Affine Subspaces

“…(3) The more general Hausdorff measure version of the Khintchine-Groshev theorem has been established in [12]. For a general background to the classical theory of metric Diophantine approximation, we refer the reader to the survey type articles [4,7].…”

Quantitative recurrence properties and homogeneous self-similar sets

“…where y stands for the distance from y to its nearest integer. Therefore, Theorem 1.3 and 1.4 correspond to a version of Khintchine's theorem and Jarník's theorem on the middle third Cantor set respectively (see [4] for more backgrounds), which provide an answer to Mahler's problem on Diophantine approximation on Cantor set [13]. One can refer to [11] for a similar answer, where the points in K are approximated by rational numbers p/q with q ∈ {3 n : n ∈ N}.…”

A general mass transference principle

“…We include one such application in Section 6 where we use Theorem 1 to deduce Hausdorff measure and dimension results for a family of random lim sup sets. Within Diophantine Approximation, it is reasonable to expect that Theorem 1 will enable the establishment of further Hausdorff measure statements relating to approximation on manifolds (see [3,4] and the references therein for more on this problem). Within Dynamical Systems, it is also reasonable to expect that Theorem 1 will allow one to study a wider class of shrinking target problems, in particular when our target is allowed to have a more exotic structure (see [20,22,23] and the references therein for more on this problem).…”