Effective equidistribution of lattice points in positive characteristic
Tal Horesh, Frédéric Paulin
TL;DR
This work establishes an effective joint equidistribution for primitive lattice points and their associated gcd solutions in the positive-characteristic setting of global function fields. By translating lattice-point questions into the representation-theoretic framework of ${ m SL}_2(K_ u)$ and applying Gorodnik–Nevo counting in well-rounded families, it achieves a quantitative equidistribution with an explicit rate $O(q_ u^{2n(- au+ ext{δ})})$ and a sharp exponent $ au\, extin(0, frac18]$. The results generalize known real/zero-characteristic phenomena to function fields and yield both primitive-vector counting with congruence constraints and a comprehensive application to continued fractions over ${f F}_q(Y)$, providing a positive-characteristic analogue of classical Dinaburg–Sinai and Linnik-type equidistribution. The work thus connects lattice-point problems, modular groups over function fields, and continued fraction dynamics in a unified, quantitative framework with potential implications for arithmetic dynamics and spectral methods in positive characteristic.
Abstract
Given a place $ω$ of a global function field $K$ over a finite field, with associated affine function ring $R_ω$ and completion $K_ω$, the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points $(a,b)\in {R_ω}^2$ in the plane ${K_ω}^2$, and for renormalized solutions to the gcd equation $ax+by=1$. The main tools are techniques of Goronik 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 $\ZZ^2$.