Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof of the positive density conjecture for integer Apollonian circle
  packings   A bounded Apollonian circle packing (ACP) is an ancient Greek construction
which is made by repeatedly inscribing circles into the triangular interstices
in a Descartes configuration of four mutually tangent circles. Remarkably, if
the original four circles have integer curvature, all of the circles in the
packing will have integer curvature as well. In this paper, we compute a lower
bound for the number $\kappa(P,X)$ of integers less than $X$ occurring as
curvatures in a bounded integer ACP $P$, and prove a conjecture of Graham,
Lagarias, Mallows, Wilkes, and Yan that the ratio $\kappa(P,X)/X$ is greater
than 0 for $X$ tending to infinity.

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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int