On a number of rational points on a convex curve

2005/03/15 by Fedor V. Petrov, Petrov, Fedor V.
Mathematics · #11H06 #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #math.MG #math.NT #msc:11H06

paper · pdf · doi:10.48550/arxiv.math/0503304

8 pages, submitted to FAA

arxiv created 2005/03/15 · arxiv updated 2009/12/01

Abstract

Let γ be a bounded convex curve on a plane. Then \sharp (γ∩ (\Z/n)²)=o(n^2/3). It streghtens the classical result of Jarn'ık (an upper estimate O(n^2/3)) and disproves a conjecture of Vershik on existence of the so-called \it universal Jarn'ık curve.

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