Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Limiting distributions in the space of 3-Lattices and non-improvability of Dirichlet’s approximation

Talk by Prof. Dr. Nimish Shah

**Date:** 10.05.21 **Time:** 15.00 - 16.15 **Room:** ETH HG G 19.2

Earlier a question of Davenport and Schmidt about non-improvability of Dirichlet’s approximation was answered by proving equidistribution of translates of non-degenerate regular curves in SL(n,R)/SL(n,Z) under the action of certain diagonal subgroups. I will speak about a joint work with D. Kleinbock, N. de Saxcé and P. Yang, where we show that equidistribution results of various flavors for the translates of lines, rather than curves, in the space of unimodular 3-lattices hold under certain precise Diophantine conditions on the lines. In particular, one can show that for almost all points on an irrational line, the Dirichlet’s approximation cannot be improved. Our proof uses Ratner’s theorem and the linearization technique combined with ideas from geometric invariant theory.