Rational Points of Definable Sets and Results of André–Oort–Manin–Mumford type

We prove some simple special cases, partly new, of results of André–Oort–Manin–Mumford type using an extension to algebraic points of bounded degree of a result of Pila–Wilkie on the density of rational points on sets definable in an o-minimal structure. The strategy follows that of a recent new proof of the Manin–Mumford conjecture by Pila–Zannier, and a proof of a special (but new) case of Pink's relative Manin–Mumford conjecture by Masser–Zannier.