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Exponential sums and lattice points. III. (English) Zbl 1065.11079
Let $C$ be a planar, closed, convex curve, made up of finitely many pieces with non-vanishing radius of curvature. The curve $C$ is said to be of class $C^{r+2}$ if the radius of curvature is (piecewise) $r$ times continuously differentiable with respect to the direction of the tangent vector. The number of lattice points within or on the dilated curve $xC$ is $\text{area}(C)x^2$ plus a remainder term $D$, which should be estimated by $$D\ll x^K(\log x)^\Lambda.$$ The classical result $K= {2\over 3}$, $\Lambda= 0$ was given by J. G. van der Corput for a $C^2$-curve. The result $K= {27\over 41}+\varepsilon$ for a $C^\infty$-curve was proved by {\it 0. Trifonov} [C. R. Acad. Bulg. Sci. 41, No. 11, 25--27 (1988; Zbl 0666.10031)]. The new estimates in the circle and divisor problems by H. Iwaniec and C. J. Mozzochi, which are based on the fundamental work of E. Bombieri and H. Iwaniec, were generalized by the author [Proc. Lond. Math. Soc., III. Ser. 60, No. 3, 471--502 (1990; Zbl 0659.10057), Corrigenda in Proc. Lond. Math. Soc., III. Ser. 66, No. 1, 70 (1993; Zbl 0937.11501) and Proc. Lond. Math. Soc., III. Ser. 68, No. 2, 264 (1994; Zbl 0937.11502)] to the case of a $C^3$-curve with the $K= {7\over 11}$, $\Lambda= {47\over 22}$. Then the author [Proc. Lond. Math. Soc. 38, 523--531 (1995; Zbl 0820.11060)] gave a further refinement of the method and proved an estimate with $K = {46\over 73}$, $\Lambda={315\over 146}$. Here it was necessary to consider integer points close to a certain plane curve, which is called the resonance curve. This problem was handled trivially. Now a careful treatment leads to the improvement $K= {131\over 208}$, $\Lambda= {18627\over 8320}$.

11P21Lattice points in specified regions
11L07Estimates on exponential sums
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