Rational Approximations to Linear Subspaces

by Nicolas de Saxcé (CNRS & Université Paris-Nord)

Monday Apr 3, 2023, 4:00 PM → 5:15 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Dirichlet's theorem in Diophantine approximation implies that for any real x, there exists a rational p/q arbitrarily close to x such that |x-p/q| < 1/q². In addition, the exponent 2 that appears in this inequality is optimal, as seen for example by taking $x=\sqrt2$. In 1967, Wolfgang Schmidt suggested a similar problem, where x is a real subspace of R^d of dimension ℓ, which one seeks to approximate by a rational subspace v. Our goal will be to obtain the optimal value of the exponent in the analogue of Dirichlet's theorem within this framework. The proof is based on a study of diagonal orbits in the space of lattices in R^d.