The theorem of Pila and Wilkie counts rational points of bounded height on the transcendental part of a definable set $X$. It follows earlier work by Jarn\'ik, Bombieri-Pila, and Pila.

If the ambient o-minimal structure is polynomially bounded, we present a bound for the number of rational points of height at most $T\ge 1$ that have distance $T^{-\lambda}$ to $X$. If the definable set contains no real semi-algebraic curve, then our bound is of the same quality as in the Pila-Wilkie Theorem. That is, their number is at most $cT^\epsilon$ where $c$ and $\lambda$ depend only on $X$ and $\epsilon$. If $X$ contains connected, real semi-algebraic curves we must omit a certain tubular of their totality to recover a $cT^\epsilon$ bound.

I will discuss these results as well as the case of a family of definable sets and some applications.