Measure Theoretic Laws for Lim Sup Sets
Measure Theoretic Laws for Lim Sup Sets
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DOI：
10.1090/memo/0846
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发表时间：
2004-01
期刊：
影响因子：
--
通讯作者：
V. Beresnevich;Detta Dickinson;S. Velani
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文献类型：
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作者：
V. Beresnevich;Detta Dickinson;S. Velani
给定一个紧致度量空间（X，d），它具有一个非原子的概率测度m和一个真实的正减函数p，我们考虑X的一个"自然"类limsup子集La（p）.经典的limsup集的'好逼近'的数字在理论的度量丢番图逼近属于这一类。我们表明，m（La（p））> 0下的“全球普遍存在”的假设和一定的m-体积和的分歧。事实上，在“局部普遍性”假设下，我们证明了La（p）具有完全测度;即m（La（p））= 1。这与数论中的经典欣钦-格罗谢夫定理的发散部分类似。此外，如果“本地无处不在”的假设是满意的，一定的f-体积和发散，然后我们能够证明，Hausdorff f-测量La（p）是无限的。一个简单的后果，这是一个下界的Hausdorff维数的La（p）和各种结果有关的尺寸和措施有关的"确切顺序"集。从本质上讲，"局部普遍性"的概念出乎意料地统一了La（p）相对于自然测度m和一般Hausdorff测度的"发散"型结果。一般框架的应用包括数论、克莱因群和有理映射。即使是经典的limsup集的"好逼近"的数字，该框架加强了经典的Hausdorff措施的结果Jarnik和开放的Duffin-Schaeffer猜想Hausdorff措施。
Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable' numbers in the theory of metric Diophantine approximation fall within this class. We show that m(La(p))>0 under a `global ubiquity' hypothesis and the divergence of a certain m--volume sum. In fact, under a `local ubiquity' hypothesis we show that La(p) has full measure; i.e. m(La(p)) =1 . This is the analogue of the divergent part of the classical Khintchine-Groshev theorem in number theory. Moreover, if the 'local ubiquity' hypothesis is satisfied and a certain f-volume sum diverges then we are able to show that the Hausdorff f--measure of La(p) is infinite. A simple consequence of this is a lower bound for the Hausdorff dimension of La(p) and various results concerning the dimension and measure of related `exact order' sets. Essentially, the notion of `local ubiquity' unexpectedly unifies `divergent' type results for La(p) with respect to the natural measure m and general Hausdorff measures. Applications of the general framework include those from number theory, Kleinian groups and rational maps. Even for the classical limsup sets of `well approximable' numbers, the framework strengthens the classical Hausdorff measure result of Jarnik and opens up the Duffin-Schaeffer conjecture for Hausdorff measures.