Algebraic zeros divisors on the projective line having small diagonals and small heights and their application to adelic dynamics

We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small $g$-heights with respect to an adelic normalized weight $g$ in arbitrary characteristic and in possibly non-separable setting, and obtain local proximity estimates between the iterations of a rational function $f\in k(z)$ of degree $>1$ and a rational function $a\in k(z)$ of degree $>0$ over a product formula field $k$ of characteristic $0$, applying this quantitative adelic equidistribution result to adelic dynamics of $f$.