Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The power law for the Buffon needle probability of the four-corner
  Cantor set   Let $C_n$ be the $n$-th generation in the construction of the middle-half
Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of
side-length $4^{-n}$. The chance that a long needle thrown at random in the
unit square will meet $K_n$ is essentially the average length of the
projections of $K_n$, also known as the Favard length of $K_n$. A classical
theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero.
It is still an open problem to determine its exact rate of decay. Until
recently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peres
and Solomyak. ($\log_* n$ is the number of times one needs to take log to
obtain a number less than 1 starting from $n$). We obtain a power law bound by
combining analytic and combinatorial ideas.

14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
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The categories of this paper are: nlin
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
11
The categories of this paper are: hep-lat