On a number of rational points on a convex curve

Details On a number of rational points on a convex curve On a number of rational points on a convex curve

2005-03-15

arxiv.org/abs/math/0503304v1

Mathematics

Number Theory

8 pages, submitted to FAA

Scientific paper

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

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