Exponential sums and lattice points. (English) Zbl 0659.10057

Huxley, M. N.

Proc. Lond. Math. Soc., III. Ser. 60, No. 3, 471-502 (1990).

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 nonzero radius of curvature, the number of lattice points is accurate to order of magnitude at most \(M^{\alpha}\) for any \(\alpha >2/3\). We show that if the radius of curvature of C is continuously differentiable, then the exponent 2/3 may be replaced by 7/11, extending the result of H. Iwaniec and C. J. Mozzochi [J. Number Theory 29, 60-93 (1988; Zbl 0644.10031)] 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.

Reviewer: M.N.Huxley

MSC:

11P21	Lattice points in specified regions
11L03	Trigonometric and exponential sums (general theory)
11P55	Applications of the Hardy-Littlewood method
11H99	Geometry of numbers
41A15	Spline approximation
41A55	Approximate quadratures

Keywords:

area inside a simple closed curve; number of lattice points; two- dimensional exponential sums; smooth arc

Citations:

Zbl 0644.10031