Exponential Sums and Lattice Points

The area A inside a simple closed curve C can be estimated graphically by drawing a square lattice of sides 1/ M . The number of lattice points inside C is approximately AM 2 . If C has continuous non‐zero radius of curvature, the number of lattice points is accurate to order of magnitude at most M α for any α < ⅔. We show that if the radius of curvature of C is continuously differentiate, then the exponent ⅔ may be replaced by7 11, extending the result of Iwaniec and Mozzochi [ 4 ] in which C was a circle. On the way we obtain results on two‐dimensional exponential sums, the average rounding error of the values of a smooth function at equally spaced arguments, and the number of lattice points close to a smooth arc.