Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Buffon's needle estimates for rational product Cantor sets   Let $S_\infty=A_\infty\times B_\infty$ be a self-similar product Cantor set
in the complex plane, defined via $S_\infty=\bigcup_{j=1}^L T_j(S_\infty)$,
where $T_j:\C\to\C$ have the form $T_j(z)=\frac1{L}z+z_j$ and
$\{z_1,...,z_L\}=A+iB$ for some $A,B\subset\rr$ with $|A|,|B|>1$ and
$|A||B|=L$. Let $S_N$ be the $L^{-N}$-neighbourhood of $S_\infty$, or
equivalently (up to constants), its $N$-th Cantor iteration. We are interested
in the asymptotic behaviour as $N\to\infty$ of the {\it Favard length} of
$S_N$, defined as the average (with respect to direction) length of its
1-dimensional projections. If the sets $A$ and $B$ are rational and have
cardinalities at most 6, then the Favard length of $S_N$ is bounded from above
by $CN^{-p/\log\log N}$ for some $p>0$. The same result holds with no
restrictions on the size of $A$ and $B$ under certain implicit conditions
concerning the generating functions of these sets. This generalizes the earlier
results of Nazarov-Perez-Volberg, {\L}aba-Zhai, and Bond-Volberg.

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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph