Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Measure theoretic laws for lim sup sets   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.

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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph