Rational points of rigid-analytic sets : a Pila–Wilkie-type theorem

We establish a rigid-analytic analog of the Pila-Wilkie counting theorem, giving subpolynomial upper bounds for the number of rational points in the transcendental part of a ‫ޑ‬ p -analytic set and the number of rational functions in a ‫ކ‬ q ((t))-analytic set.For ‫((ޚ‬t))-analytic sets, we prove such bounds uniformly for the specialization to every nonarchimedean local field.