On metric diophantine approximation in matrices and Lie groups

Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, Nicolas de Saxcé

Published 2014-10-15Version 1

We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over the rationals. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.