Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.11714 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909324273516544 |
|---|---|
| author | Gaudron, Éric |
| author_facet | Gaudron, Éric |
| contents | 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 $α\in E\otimes_{K}K_{v}$ such that $q(α)=1$. 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 $\vert q(α\upphi-\upupsilon)\vert_{v}\le c\vert\upphi\vert_{v}/T$. 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{é}mond (2017). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11714 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adelic approximation on spheres Gaudron, Éric Number Theory 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 $α\in E\otimes_{K}K_{v}$ such that $q(α)=1$. 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 $\vert q(α\upphi-\upupsilon)\vert_{v}\le c\vert\upphi\vert_{v}/T$. 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{é}mond (2017). |
| title | Adelic approximation on spheres |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.11714 |