Given a compact metric space (Ω, d) equipped with a non-atomic, probability measure m and a positive decreasing function ψ, we consider a natural class of lim sup subsets Λ(ψ) of Ω. The classical lim sup set W (ψ) of 'ψ-approximable' numbers in the theory of metric Diophantine approximation fall within this class. We establish sufficient conditions (which are also necessary under some natural assumptions) for the m-measure of Λ(ψ) to be either positive or full in Ω and for the Hausdorff fmeasure to be infinite. The classical theorems of Khintchine-Groshev and Jarník concerning W (ψ) fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantine approximation including those for real, complex and p-adic fields associated with both independent and dependent quantities. Applications also include those to Kleinian groups and rational maps.Compared to previous works our framework allows us to successfully remove many unnecessary conditions and strengthen fundamental results such as Jarník's theorem and the Baker-Schmidt theorem. In particular, the strengthening of Jarník's theorem opens up the Duffin-Schaeffer conjecture for Hausdorff measures. MathematicsSubject Classification: 11J83; 11J13, 11K60, 28A78, 28A80 §9.1. The subset A(ψ, B) of Λ(ψ) ∩ B 30 §9.2. Proof of Lemma 8 : quasi-independence on average 34 Section 10. Proof of Theorem 2: 0 ≤ G < ∞ 37 §10.1. Preliminaries 37 §10.2. The Cantor set Kη 40 §10.3. A measure on Kη 52 Section 11. Proof of Theorem 2: G = ∞ 60 §11.1. The Cantor set K and the measure µ 61 §11.2. Completion of the proof 62 Section 12. Applications 64 §12.1. Linear Forms 64 §12.2. Algebraic Numbers 66 §12.3. Kleinian Groups 68 §12.4. Rational Maps 74 §12.5. Diophantine approximation with restrictions 78 §12.6. Diophantine approximation in Qp 79 §12.7. Diophantine approximation on manifolds 81 §12.8. Sets of exact order 86 Bibliography 89 1. INTRODUCTION Jarník's Theorem (1931). Let f be a dimension function such that r −1 f (r) → ∞ as r → 0 and r −1 f (r) is decreasing. Let ψ be a real, positive decreasing function.ThenClearly the above theorem can be regarded as the Hausdorff measure version of Khintchine's theorem. As with the latter, the divergence part constitutes the main substance. Notice, that the case when H f is comparable to one-dimensional Lebesgue measure m (i.e. f (r) = r) is excluded by the condition r −1 f (r) → ∞ as r → 0 . Analogous to Khintchine's original statement, in Jarník's original statement the additional hypotheses that r 2 ψ(r) is decreasing, r 2 ψ(r) → 0 as r → ∞ and that r 2 f (ψ(r)) is decreasing were assumed. Thus, even in the simple case when f (r) = r s (s ≥ 0) and the approximating function is given by ψ(r) = r −τ log r (τ > 2), Jarník's original statement gives no information regarding the s-dimensional Hausdorff measure of W (ψ) at the critical exponent s = 2/τ -see below. That this is the case is due to the fact that r 2 f (ψ(r)) is not decreasing. However, as we shall see these additional hypotheses ...

A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

Let A n,m (ψ) denote the set of ψ-approximable points in R mn . Under the assumption that the approximating function ψ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A n,m (ψ). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on ψ is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n ≥ 3 (Schmidt) or when n = 1 and m ≥ 2 (Gallagher). Surprisingly, when n = 2 the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multi-dimensional analogue of Catlin's Conjecture.

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of R n . These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarník type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifoldsa very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions n > 2 and show that they are ubiquitous (that is uniformly distributed).