Lattice points on a curve via $\ell^2$ decoupling
We prove a generalization of Bombieri-Pila's inequality for lattice points on a curve to proper generalized arithmetic progressions. The proof uses the $\ell^2$ decoupling inequality for non-degenerate curves in $\mathbb{R}^n$. We also review the curve-lifting method and use it to establish various estimates of lattice points on a planar curve.