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SUMMARY:Attaching shortest vectors to lattice points and applications - An
 \, J (Peking University)
DTSTART:20140702T080000Z
DTEND:20140702T085000Z
UID:TALK53281@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We highlight a simple construction\, appeared in the work of D
 . Badziahin\, A. Pollington and S. Velani where they proved Schmidt's conj
 ecture\, which attaches to a lattice point an integral vector that is shor
 test in a certain sense. Such a construction turns out to be useful in stu
 dying badly approximable vectors and bounded orbits of unimodular lattices
 . It can be used to prove: (1) The set $mathrm{Bad}(i\,j)$ of two-dimensio
 nal badly approximable vectors is winning for Schmidt's game\; (2) $mathrm
 {Bad}(i\,j)$ is also winning on non-degenerate curves and certain straight
  lines\; (3) Three-dimensional unimodular lattices with bounded orbits und
 er a diagonalizable one-parameter subgroup form a winning set (at least in
  a local sense).\n
LOCATION:Seminar Room 1\, Newton Institute
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