Exponential Sums and Lattice Points Ii

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. If C has continuous non-zero radius of curvature, then the number of lattice points is accurate to order of magnitude at most M for any a> §. We show that if the radius of curvature of C is continuously differentiate, then the exponent § may be replaced by 75, improving the exponent -ft of Iwaniec and Mozzochi [15] (for whom C was a circle) and the author [8]. We use results on two-dimensional exponential sums and rounding error sums. Assuming further differentiability, we obtain a stronger result in the mean for a family of lattice point problems. Applications to quadrature are given.