Effective equidistribution of lattice points in positive characteristic

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 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 ${\text{ZZ}}^2$.