Measure theoretic laws for lim sup sets

Abstract

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 ...

Ubiquity and large intersections properties under digit frequencies constraints

“…Similar results were obtained in R d for various families {(x n , l n )} n≥1 [8,9,1,23] or in metric spaces enjoying enough self-similar properties to support a monofractal measure [13,14,4], like the middle third Cantor set [26].…”

On weighted inhomogeneous Diophantine approximation on planar curves

“…For s = 1 the following is its generalization in terms of general Hausdorff measure and a consequence of Corollary 3 in [5].…”

On a Problem of Bernik, Kleinbock and Margulis

“…In the case n = 1, it was first described in 1924 by Khintchine [23] in his beautiful discovery of a zero-one law stated below in a slightly stronger modern version, see [8].…”

“…Following Beresnevich, Dickinson and Velani [8], we will say that a function f is 2-regular if there is a positive λ < 1 such that f (2 t+1 ) ≤ λf (2 t ) for all sufficiently large t. Also, we will say that f is quasi-monotonic if there are constants z and c 1 such that 0 < z < 1 ≤ c 1 and f (zx) ≤ c 1 f (x) for all sufficiently large x. THEOREM 1. Let 1 , 2 : ‫ޒ‬ + → ‫ޒ‬ + be such that 1 2 is a monotonically decreasing function.…”

“…Let G J be the set arising from Lemma 1 and x ∈ G J . Then, by Lemma 1, there is a primitive irreducible polynomial P ∈ ‫[ޚ‬x] of degree n satisfying (8).…”