Rational points of bounded height on entire curves

Let $X$ be an affine or a projective variety defined over a number field $K$ and $\phi :\mathbf{C}\to X(\mathbf{C})$ be a holomorphic map with Zariski dense image. We estimate the number of rational points of height bounded by $H$ in the image of a disk of radius $r$ in terms of the the Nevanlinna characteristic function of $\phi$ and $H$ in a way which generalize the classical Bombieri–Pila estimate to expanding domains. In general this bound is exponential but we show that for many values of $H$ and $r$, the bound is polynomial.