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Exponential sums and lattice points. (English) Zbl 0659.10057
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
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