Measure Theoretic Laws for Lim Sup Sets

Victor Beresnevich, Detta Dickinson, Sanju Velani - Collection Memoirs of the American Mathematical Society

92 pages, parution le 29/06/2006

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Résumé

Given a compact metric space $(\Omega,d)$ equipped with a non-atomic, probability measure $m$ and a positive decreasing function $\psi$, we consider a natural class of lim sup subsets $\Lambda(\psi)$ of $\Omega$. The classical lim sup set $W(\psi)$ of `$\p$-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 $\Lambda(\psi)$ to be either positive or full in $\Omega$ and for the Hausdorff $f$-measure to be infinite. The classical theorems of Khintchine-Groshev and Jarník concerning $W(\psi)$ 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.

Sommaire

Introduction
Ubiquity and conditions on the general setp
The statements of the main theorems
Remarks and corollaries to Theorem 1
Remarks and corollaries to Theorem 2
The classical results
Hausdorff measures and dimension
Positive and full $m$-measure sets
Proof of Theorem 1
Proof of Theorem 2: $0\leq G < \infty$
Proof of Theorem 2: $G= \infty$
Applications
Bibliography

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	PAPIER
Éditeur(s)	American Mathematical Society (AMS)
Auteur(s)	Victor Beresnevich, Detta Dickinson, Sanju Velani
Collection	Memoirs of the American Mathematical Society
Parution	29/06/2006
Nb. de pages	92
Format	17,5 x 25
Couverture	Broché
Poids	223g
Intérieur	Noir et Blanc
EAN13	9780821838273
ISBN13	978-0-8218-3827-3