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.

Let K denote the middle third Cantor set and A := {3 n : n = 0, 1, 2, . . .}. Given a real, positive function ψ let W A (ψ) denote the set of real numbers x in the unit interval for which there exist infinitely many (p, q) ∈ Z × A such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for W A (ψ) ∩ K. One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K -an assertion attributed to K. Mahler.

Summary. To any dynamical system equipped with a metric, we associate a class of "well approximable" sets. In the case of an expanding rational map of the Riemann sphere acting on its Julia set, we estimate and in some cases compute the Hausdorff dimension of the associated "well approximabie" sets. The methods used show a clear link between distortion properties and the type of results obtained in this paper, via ergodic theory and ubiquity. Motivation and statements of the resultsConsider a metric space J equipped with a Borel probability measure/t. If T : J ~ J is measure preserving and ergodic, one knows by the Ergodic Theorem that for any ball B C J of positive measure, the subset {z E J : Tn(z) E B for infinitely many n E IN} of J has full Ft-measure. This means that the trajectories of almost all points will go through the ball B infinitely often. In general one can ask the question what happens if the ball B shrinks with time. More precisely if at time n one has a ball B(n) = B(zo, r(n)) centred at a point Zo E J of radius r(n) (r(n) ~ 0 as n --~ oo), then what kind of properties does the set of points z have, whose images Tn(z) are in B(n) for infinitely many n?These points can be thought of as trajectories which hit a shrinking target infinitely often. We shall call such points "well approximable" in analogy with the classical theory of metric Diophantine approximation [4,18] and its more recent extensions to the theory of discrete hyperbolic groups (see for example [7,13,21]). In the classical theory, the projective real line, F, U {cx~} is identified with the unit tangent space at a point of the modular surface IH/SL~(Z) (the modular group SL2(77) acts on the upper half plane model

We study the Patterson measure on the limit set of a geometrically finite Kleinian group with parabolic elements. Using the 5-conformality of the Patterson measure in combination with basic hyperbolic geometry we obtain a global Shadow Lemma. This macroscopic measure estimate is exploited to derive a Khintchine-type theorem, which is then used for a finer analysis of the density of the limit set in terms of Hausdorff measure.Proof. Fix an element q in P. Lemma 2.1 and the fact that a q lies in H(L(G)) c 26 r s gf ji(b(r lt ))Applying Lemma 3.3 to estimate the \x-measure of £(17,), we then obtain Clearly d{i\ ti G(0)) = d(g(r) t ), G(0)), whence by the triangle inequality,220 THE PATTERSON MEASURE PROPOSITION 4.11. For all positive e, H^LiG)) = °°.Proof. Proposition 4.10 implies that, for any positive £, for /^-almost every £ in E e and for all positive r^2e~' c , This, combined with the above first general remark on Hausdorff measures, implies for all positive e, that H^e(E E )^k2ifJi(E e )>0. Let £] and e 2 denote two positive numbers such that e, < e 2 -Then lim x _ 0 {4> e J

c (£ fl ) = 0. This is a contradiction, whence the proposition follows.

The convergence theory for the set of simultaneously $\psi$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the distribution of rational points' (by V. Beresnevich, H. Dickinson, and S. Velani, with an Appendix: `Sums of two squares near perfect squares' by R.C. Vaughan, To appear: Annals of Math., Pre-print: arkiv:math.NT/0401148, (2004), 1-52.) and thereby completes the general metric theory for planar curves.Comment: 18 pages. To appear in Invent. Mat