On a generalization of a theorem of Popov

TL;DR

This paper extends Popov's theorem by providing sharp lattice point estimates near dilated parabolas using advanced analytic number theory techniques.

Contribution

It generalizes Popov's old result by deriving precise bounds for lattice points near general parabolas through novel analytical methods.

Findings

Sharp estimates for lattice points under and near dilated parabolas

Generalization of Popov's theorem with improved bounds

Application of Vaaler's lemma and Erdős-Turán inequality

Abstract

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 Popov. We apply Vaaler's lemma and the Erd\H{o}s-Turan 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.