Abstract

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 $q:E\to K$ 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 $(\upsilon ,\varphi )\in E\times K$, with $\max <!--FUNCTION\; APPLICATION-->(\parallel \upsilon {\parallel }_{E,v},|\varphi {|}_v)\le T$ and $\max <!--FUNCTION\; APPLICATION-->(\parallel \upsilon {\parallel }_{E,w},|\varphi {|}_w)$ controlled for any place $w$, satisfying $q(\upsilon )={\varphi }^2\ne 0$ and $|q(\alpha \varphi -\upsilon ){|}_v\le c|\varphi {|}_v∕T$. This remains true for certain infinite algebraic extensions as well as for a compact set of places of $K$. Our statements generalize and improve on earlier results by Kleinbock and Merrill (2015) and Moshchevitin (2017). The proofs rely on the quadratic Siegel’s lemma in a rigid adelic space obtained by the author and Rémond (2017).

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