The Bombieri-Pila theorem predicts a bound of the form $c(f,\epsilon)H^\epsilon$ for the number of rational points of height at most $H$ on the graph of a (real analytic) transcendental function $f$ restricted to a compact interval. Although this bound is sharp in general, for certain special cases (such as those arising under additional hypotheses on $f$) it can be improved to a poly-logarithmic bound in $H$. In our recent work we obtain a $C(\log H)^\eta$ bound for the number of algebraic points of bounded degree and height at most $H$ on certain subsets of graphs of a family of entire functions defined by a canonical product.