The number of lattice points on a convex curve

Abstract

If C is a strictly convex plane curve of length l, it has been known for a long time that the number of integer lattice points on C is O(l 2 3 ) and the exponent is best possible. In this paper, it is shown that the exponent can be decreased by imposing suitable smoothness conditions on C; in particular, if C has a continuous third derivative with a sensible bound, the best possible value of the exponent lies between 3 5 and 1 3 inclusive.