Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve. Does it generalize to a bound for the count of lattice points near (say within a distance of $\delta$) an algebraic curve? Is it obvious that we only need to add $O(\delta L)$ to the Bombieri-Pila count, where $L$ is the length of the curve? Or do we have to use the weaker bound of Swinnerton-Dyer?