Effective equidistribution of lattice points 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)∈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 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 ℤ 2 .