Diophantine approximation on spheres

Dirichlet’s approximation theorem on spheres consists in approaching a vector located on the unit n-sphere by another vector on this sphere having rational coordinates. We explain how to get such a result in general, even when the underlying quadratic form is no longer positive-definite. Our statements generalize and improve on earlier results by Kleinbock & Merrill (2015) and Moshchevitin (2017). The proofs rely on the quadratic Siegel’s lemma obtained by the author and Gaël Rémond (2017)