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On a generalization of a theorem of Popov. (English) Zbl 1456.11190

Summary: In this paper, we obtain sharp estimates for the number of lattice points under and near the dilation of a general parabola, the former generalizing an old result of V. N. Popov [Math. Notes 18, 1007–1010 (1976; Zbl 0327.10049)]. We apply Vaaler’s lemma [J. D. Vaaler, Bull. Am. Math. Soc., New Ser. 12, 183–216 (1985; Zbl 0575.42003)] and the Erdős-Turán inequality to reduce the two underlying counting problems to mean values of a certain quadratic exponential sums, whose treatment is subject to classical analytic techniques.

MSC:

11P21 Lattice points in specified regions
11J25 Diophantine inequalities
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References:

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