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 $q\colon E\to K$ be a quadratic form. Let $v$ be a place of $K$ and $\alpha\in E\otimes_{K}K_{v}$ such that $q(\alpha)=1$. We produce an explicit constant $c$ having the following property. If there exists $x\in E$ such that $q(x)=1$ then, for any $T>c$, there exists $(\upupsilon,\upphi)\in E\times K$, with $\max{(\Vert\upupsilon\Vert_{E,v},\vert\upphi\vert_{v})}\le T$ and $\max{(\Vert\upupsilon\Vert_{E,w},\vert\upphi\vert_{w})}$ controlled for any place $w$, satisfying $q(\upupsilon)=\upphi^{2}\ne 0$ and $\vert q(\alpha\upphi-\upupsilon)\vert_{v}\le c\vert\upphi\vert_{v}/T$. 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).