Skip to content

GAN

Seminar

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

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 $\mathrm{Bad}(i,j)$ 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).

Video

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.

Back to top ∧