Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher finiteness properties of arithmetic approximate lattices: The
  Rank Theorem for number fields   We introduce geometric and homological finiteness properties for countable
approximate groups via coarse geometry and then study these finiteness
properties for S-arithmetic reductive approximate groups. For S-arithmetic
approximate groups without infinite places we show that the finiteness length
is finite and compute this finiteness length explicitly. In the simple case it
is one less than the sum of the local ranks. This extends the Rank Theorem of
Bux, K\"ohl and the second author from positive characteristic to
characteristic zero. Our proof is based on a geometric version of their proof,
but except for some input from reduction theory it is characteristic free. This
indicates that the apparent differences between arithmetic groups in
characteristic zero and positive characteristic concerning finiteness
properties are entirely due to the presence of infinite places.

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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
1
The categories of this paper are: cs
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio