Attaching shortest vectors to lattice points and applications
Duration: 53 mins 54 secs
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About this item
| Description: |
An, J (Peking University)
Wednesday 02 July 2014, 09:00-09:50 |
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| Created: | 2014-07-11 18:30 |
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| Collection: | Interactions between Dynamics of Group Actions and Number Theory |
| Publisher: | Isaac Newton Institute |
| Copyright: | An, J |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | We highlight a simple construction, appeared in the work of D. Badziahin, A. Pollington and S. Velani where they proved Schmidt's conjecture, which attaches to a lattice point an integral vector that is shortest in a certain sense. Such a construction turns out to be useful in studying badly approximable vectors and bounded orbits of unimodular lattices. It can be used to prove: (1) The set Bad(i,j) of two-dimensional badly approximable vectors is winning for Schmidt's game; (2) Bad(i,j) is also winning on non-degenerate curves and certain straight lines; (3) Three-dimensional unimodular lattices with bounded orbits under a diagonalizable one-parameter subgroup form a winning set (at least in a local sense). |
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