Adelic approximation on spheres

We establish an adelic version of Dirichlet's approximation theorem on spheres. Let K be a number field, E be a rigid adelic space over K and q E→ K be a quadratic form. Let v be a place of K and α∈ E⊗KKᵥ such that q(α)=1. We produce an explicit constant c having the following property. If there exists x∈ E such that $q(x)=1$ then, for any $T>c$, there exists (,)∈ E× K, with max(E,v,ᵥ)≤ T and max(E,w,w) controlled for any place w, satisfying q()=²≠ 0 and q(α-)ᵥ≤ cᵥ/T. This remains true for some infinite algebraic extensions as well as for a compact set of places of K. Our statements generalize and improve on earlier results by Kleinbock \& Merrill (2015) and Moshchevitin (2017). The proofs rely on the quadratic Siegel's lemma in a rigid adelic space obtained by the author and R{\'e}mond (2017).