Let ψ : R + → R + be a non-increasing function. A real number x is said to be ψ-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system |qx − p| < ψ(t) and |q| < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(ψ). In this paper we prove that the Hausdorff measure of the complement D(ψ) c (the set of ψ-Dirichlet non-improvable numbers) obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock & Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

Abstract. In this paper we investigate the metrical theory of Diophantine approximation associated with linear forms that are simultaneously small for infinitely many integer vectors; i.e. forms which are close to the origin. A complete Khintchine-Groshev type theorem is established, as well as its Hausdorff measure generalization. The latter implies the complete Hausdorff dimension theory.

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine-Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximating function.2000 Mathematics Subject Classification. 11J83.

Let Ψ : [1, ∞) → R + be a non-decreasing function, a n

The well known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.1.1. Dual Diophantine approximation. To set the scene for the problems considered in this paper, we first recall the fundamental results in the theory of dual Diophantine approximation. Let ψ : R + → R + denote a real positive decreasing function. We refer to ψ as an approximating function. Define the set W (ψ) := x = (x 1 , . . . , x n ) ∈ R n : |a 0 + a 1 x 1 + · · · + a n x n | < ψ(|a|)where 'i.m.' stands for 'infinitely many' and |a| = max{|a 1 |, . . . , |a n |} is the standard supremum norm. A vector x ∈ W (ψ) will be called ψ−approximable. In the case ψ(r) = r −τ for some τ > 0 we also say that x is τ −approximable and denote W (ψ) by W (τ ). The first significant result in the theory is Dirichlet's theorem which tells us that W (n) = R n .The following fundamental result provides a beautiful criterion for the 'size' of the set W (ψ) in terms of n-dimensional Lebesgue measure | · | n .