Measure Theoretic Laws for Lim Sup Sets

Victor Beresnevich, Detta Dickinson, Sanju Velani (Autoren)

Buch | Softcover

91 Seiten

2005
American Mathematical Society (Verlag)
978-0-8218-3827-3 (ISBN)

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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 Jarnik 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 Jarnik's theorem and the Baker-Schmidt theorem. In particular, the strengthening of Jarnik's theorem opens up the Duffin-Schaeffer conjecture for Hausdorff measures.

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.

Erscheint lt. Verlag	31.12.2006
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Gewicht	227 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
ISBN-10	0-8218-3827-X / 082183827X
ISBN-13	978-0-8218-3827-3 / 9780821838273
Zustand	Neuware