The area within a curve
Let C be a simple closed convex and smooth curve with continuous non-zero radius of curvature and area A. For a large positive variable R consider the blow up domain by the factor R such that it has the area \(AR^ 2\). If N denotes the number of lattice points within it, then it is well- known that \[ (1)\quad N=AR^ 2+O(R^{2/3+\epsilon})\quad (\epsilon >0). \] Recently \textit{H. Iwaniec} and \textit{C. J. Mozzochi} [J. Number Theory 29, No.1, 60-93 (1988; Zbl 0644.10031)] obtained the estimation \(R^{7/11+\epsilon}\) for the error term. The author and \textit{N. Watt} [Proc. Lond. Math. Soc., III. Ser. 57, 1-24 (1988; Zbl 0644.10027); Exponential sums and lattice points (to appear)] showed that their method can be generalized to more general domains. In this paper the author gives a strictly elementary proof of (1) using only the continued fraction algorithm and the circle method. This gives an interesting new view on the work of H. Iwaniec and C. J. Mozzochi. Also simple estimates for the mean square error are obtained.
scientific article; zbMATH DE number 4071033