The number of integral points on arcs and ovals

Abstract

integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed, C, strictly convex arc Γ and consider the number of lattice points on tΓ, the dilation of Γ by a factor t, t ≥ 1. This of course is the same as asking for rational points (mN , n N ) on Γ as N → ∞. In fact, Swinnerton–Dyer proves a bound of type |tΓ ∩ ZZ| ≤ c(Γ, e)t 3 5+e