Seminar at SMU Delhi

In 1926, the Czech mathematician J\’{a}rnik proved that a strictly convex curve of length $\ell$ in $\mathbb{R}^2$ contains at most $c (\ell)^{2/3}$ “integral” points, i.e. points in $\mathbb{Z}^2$, and showed that this result is optimal. \\ In 1988, G. Grekos dealt with the case of “flat” curves (*). The case of “very flat” curves{**} has been dealt with by G. Grekos, J-M. Deshouillers and A. Urbis (2016). \\ The talk aims to address a non specialized audience.\\ (*) i.e. $\mathcal{C}^2$ curves for which the length is between $r^{2/3}$ and $r$, where $r$ is the minimal value of the radius of the curve.\\ (**} i.e. $\mathcal{C}^2$ curves for which the length is less than $r^{2/3}$.\\