Effective equidistribution of lattice points in positive characteristic

Horesh, Tal; Paulin, Frédéric

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 679-703.

Étant donné une place $\omega$ d’un corps de fonctions global $K$ sur un corps fini, d’anneau des fonctions affines associé $R_{\omega }$ et de complétion $K_{\omega }$, le but de ce texte est de donner un résultat d’équidistribution jointe effectif pour les points entiers primitifs renormalisés $(a,b)\in {R_{\omega }}^2$ du plan ${K_{\omega }}^2$, et pour les solutions renormalisées de l’équation du pgcd $ax+by=1$. Les outils principaux sont les techniques de Gorodnik et Nevo sur le comptage de points entiers dans des familles de parties bien arrondies. Ceci donne un résultat plus précis en caractéristique positive d’un résultat de Nevo et du premier auteur sur l’équidistribution des points entiers primitifs de ${\mathbb{Z}}^2$.

Given a place $\omega$ of a global function field $K$ over a finite field, with associated affine function ring $R_{\omega }$ and completion $K_{\omega }$, the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points $(a,b)\in {R_{\omega }}^2$ in the plane ${K_{\omega }}^2$, and for renormalized solutions to the gcd equation $ax+by=1$. The main tools are techniques of Gorodnik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in ${\mathbb{Z}}^2$.

Classification : 11J70, 11N45, 14G17, 20G30, 11K50, 28C10, 11P21
Mots clés : lattice point, equidistribution, positive characteristic, function fields, continued fraction expansion

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