Trigonometric Polynomials and Lattice Points

descriptionPublicationkeyboard_double_arrow_right Article , Other literature type 01 Jan 1992Publisher:JSTORJournal:Proceedings of the American Mathematical Society, volume 115, page 899 (issn: 0002-9939,

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Authors: Cilleruelo, J.; Cordoba, A.;

doi: 10.2307/2159332 , 10.1090/s0002-9939-1992-1089403-8

handle: 10261/31179

Trigonometric Polynomials and Lattice Points

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Abstract

In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius R R , an arc whose length is smaller than 2 R 1 / 2 − 1 ( 4 [ m / 2 ] + 2 ) \sqrt 2 {R^{1/2 - 1(4[m/2] + 2)}} contains, at most, m m lattice points. We use the same method to obtain sharp L 4 {L^4} -estimates for uncompleted, Gaussian sums

Keywords

incomplete Gaussian sums, Lattice points in specified regions, precise \(L^ 4\)-asymptotic, circles, distribution of lattice points, Gauss and Kloosterman sums; generalizations

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