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?

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance of $\delta=o(1)$) 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?

Edit: To put things in more context, $O(\delta L)$ is the "expected" number of lattice points near the curve if the curve is situated randomly in the plane. The Bombieri-Pila bound can be thought of as arising from the arithmetic structure of the algebraic curve. My question then amounts to: do the expected count and the arithmetic count together fully account for the lattice points near an algebraic curve?