  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}.
