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 ...

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.

Abstract. The Hausdorff dimension and measure of the set of simultaneously ψ-approximable points lying on integer polynomial curves is obtained for sufficiently small error functions. §1. Introduction and notation. In dimensions higher than one there are two standard forms of Diophantine approximation and they have rather different properties. To describe these ideas some notation and terminology is needed. For each t ∈ R letThe supremum norm will be denoted by | . |, that is, for a vector x ∈ Z n , |x| = max{|x 1 |, . . . , |x n |}.Throughout this paper, hcf(x, y) will be used to denote the highest common factor of the integers x and y.Let ψ be a decreasing function such that ψ(r ) → 0 as r → ∞. The set S ψ (M) of simultaneously ψ-approximable points lying on an m-dimensional manifold M embedded in R n is defined byThere is a natural dual to this set, namelyIf ψ(r ) = r −τ then the sets are denoted S τ (M) and L τ (M), respectively. Obviously any element of Q n lying on M is in S ψ (M) for all functions ψ. Correspondingly, the intersection of M with a rational hyperplane given by the equation q.x = p (for p ∈ Z and q ∈ Z m ) is contained in L ψ (M) for all ψ. Any other points in either S ψ (M) or L ψ (M) lie "close" to these points or planes. In this paper we will study S ψ (M) when M is a polynomial curve and ψ(r ) → 0 sufficiently fast.

For each real number α, let E(α) denote the set of real numbers with exact order α. A theorem of Güting states that for α ≥ 2 the Hausdorff dimension of E(α) is equal to 2/α. In this note we introduce the notion of exact t-logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Background and statement of results The classical theoryFor each real number τ , let W (τ ) denote the set of real numbers which are τ -well approximable, that is W (τ ) := {x ∈ R : |x − p/q| ≤ |q| −τ for infinitely many rationals p/q} . For a real number x, its exact order τ (x) is defined as follows:τ (x) := sup{τ : x ∈ W (τ )} .It follows from Dirichlet's theorem in the theory of Diophantine approximation that τ (x) ≥ 2 for all x ∈ R. For α ≥ 2, let E(α) denote the set of numbers with exact order α; that is E(α) := {x ∈ R : τ (x) = α} . Royal Society University Research Fellow 254 V. Beresnevich et al.The set E(α) is equivalent to the set of real numbers x for which Mahler's function θ 1 (x) is equal to α − 1; the general function θ n (x) is central to Mahler's classification of transcendental numbers (see [1,4]). In [4], Güting proved the following result:Throughout, dim X will denote the Hausdorff dimension of the set X. For further details regarding Hausdorff dimension and measure see Sect. 1.3. At this point it is worth mentioning the classical result of Jarník and Besicovitch which states that dim W (α) = 2/α. Thus, Güting's result says that the 'size' of the sets E(α) and W (α) expressed in terms of Hausdorff dimension are the same. In this note we give a short proof of Güting's theorem and show that the s-dimensional Hausdorff measure of the set E(α) at the critical exponent is infinity, that isThese exact order results are a simple consequence of our results on exact tlogarithmic order. Linear forms and exact t-logarithmic orderLet ψ : R + → R + be a real positive function. An m×n matrix X = (x ij ) ∈ R mn is said to be ψ-well approximable if the system of inequalitiesis satisfied for infinitely many vectors q ∈ Z m , p ∈ Z n . Here |q| denotes the supremum norm of the vector q ; i.e. |q| = max{|q 1 | , . . . , |q m |} . The systemof n real linear forms in m variables q 1 , . . . , q m will be written more concisely as q X ,where the matrix X is regarded as a point in R mn . In view of this notation, the set of ψ-well approximable points will be denoted bywhere 'i.m.' means 'infinitely many'. By definition, |qX − p| = max 1≤j ≤n |q.X (j ) − p j | where X (j ) is the j'th column vector of X. In the case when ψ(r) = r −τ we denote W (m, n; ψ) by W (m, n; τ ) and note that when m = n = 1 the ...

This paper concerns perturbations of smooth vector fields on T n (constant if n 3) with zeroth-order C ∞ and Gevrey G σ , σ 1, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the C ∞ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in C ∞ and Gevrey G σ . Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the 'size' of different exceptional sets, including some inhomogeneous examples.