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Attaching shortest vectors to lattice points and applications

An, J (Peking University)
Wednesday 02 July 2014, 09:00-09:50

Seminar Room 1, Newton Institute


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 $\mathrm{Bad}(i,j)$ of two-dimensional 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 under a diagonalizable one-parameter subgroup form a winning set (at least in a local sense).


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