On a number of rational points on a convex curve
Abstract
Description
Let $γ$ be a bounded convex curve on a plane. Then $\sharp (γ\cap (\Z/n)^2)=o(n^{2/3})$. 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
8 pages, submitted to FAA