  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 $f$.
