Measure Theoretic Laws for Lim Sup Sets
| Author/creator | Beresnevich, Victor, 1971- Author |
| Other author | Dickinson, Detta 1968- Author |
| Other author | Velani, Sanju 1966- Author |
| Format | Electronic |
| Publication Info | Providence : American Mathematical Society |
| Description | 91 p. |
| Supplemental Content | Full text available from Ebook Central - Academic Complete |
| Supplemental Content | Full text available from Memoirs of the American Mathematical Society - Backfile |
| Subjects |
| Series | Memoirs of the American Mathematical Society Ser. 179 |
| Summary | Annotation 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 Jarn©Ưk concerning $W(psi)$ fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantineapproximation 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 Jarn©Ưk's theorem and the Baker-Schmidt theorem. In particular, the strengthening of Jarn©Ưk's theorem opens up the Duffin-Schaeffer conjecturefor Hausdorff measures. |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Genre/form | Electronic books. |
| LCCN | 2005053661 |
| ISBN | 9780821838273 |
| ISBN | 082183827X (Trade Paper) Active Record |
| Standard identifier# | 9780821838273 |
| Stock number | MEMO/179/846 00001436 |
Availability
| Library | Location | Call Number | Status | Item Actions |
|---|---|---|---|---|
| Electronic Resources | ✔ Available |