TY - JOUR
AU - Huxley, M. N.
AB - 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 AM2. If C has continuous non‐zero radius of curvature, then the number of lattice points is accurate to order of magnitude at most Mα for any α> ⅔. We show that if the radius of curvature of C is continuously differentiate, then the exponent ⅔ may be replaced by 4673, improving the exponent 711 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.
TI - Exponential Sums and Lattice Points II
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-66.2.279
DA - 1993-03-01
UR - https://www.deepdyve.com/lp/wiley/exponential-sums-and-lattice-points-ii-Tmd3CRBglS
SP - 279
EP - 301
VL - s3-66
IS - 2
DP - DeepDyve
ER -