On a number of rational points on a convex curve

Abstract

Let $\gamma$ be a bounded convex curve on a plane. Then $♯(\gamma \cap (\mathbb{Z}/n{)}^2)=o(n^{2/3})$. 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}.

8 pages, submitted to FAA