$p$-adic Directions of Primitive Vectors

Guilloux, Antonin; Horesh, Tal

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 85-107

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive ${\mathbb{Z}}^2$ points in the $p$-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple $S$-arithmetic groups.

Les problèmes de type Linnik concernent la distribution des projections des points entiers sur la sphère unitaire lorsque leur norme augmente et différentes généralisations de ce phénomène. Notre travail s’intéresse à une question de ce type : nous prouvons la distribution uniforme des projections des points primitifs de ${\mathbb{Z}}^2$ sur la sphère unitaire $p$-adique lorsque leur norme (réelle) tend vers l’infini. La preuve se fait en comptant les points d’un réseau dans des $S$-groupes arithmétiques semi-simples.

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