Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diophantine approximations for translation surfaces and planar resonant
  sets   We consider Teichm\"uller geodesics in strata of translation surfaces. We
prove lower and upper bounds for the Hausdorff dimension of the set of
parameters generating a geodesic bounded in some compact part of the stratum.
Then we compute the dimension of those parameters generating geodesics that
make excursions to infinity at a prescribed rate. Finally we compute the
dimension of the set of directions in a rational billiard having fast
recurrence, which corresponds to a dynamical version of a classical result of
Jarn\'ik and Besicovich. Our main tool are planar resonant sets arising from a
given translation surface, that is the countable set of directions of its
saddle connections or of its closed geodesics, filtered according to length. In
an abstract setting, and assuming specific metric properties on a general
planar resonant set, we prove a dichotomy for the Hausdorff measure of the set
of directions which are well approximable by directions in the resonant set,
and we give an estimate on the dimension of the set of badly approximable
directions. Then we prove that the resonant sets arising from a translation
surface satisfy the required metric properties.

16
The categories of this paper are: stat
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat