Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase transitions in infinitely generated groups, and related problems
  in additive number theory   Let A be an infinite set of generators for a group G, and let L_A(r) denote
the number of elements of G whose word length with respect to A is exactly r.
The purpose of this note is to determine all growth functions L_A(r) associated
to infinite generating sets for groups, and to describe a phase transition
phenomenon associated with infinite generating sets. A list of open problems is
also.included.

25
The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
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The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex