Alladi Ramakrishnan Hall

Integral points on a convex curve

Jean-Marc Deshouillers

Institut math\'ematique de Bordeaux, Bordeaux INP

Let $\Gamma$ be a $\mathcal{C}^2$ strictly convex curve in the euclidean plane $\mathbb{R}^2$. We denote by $\ell = \ell(\Gamma)$ its length and $N = N(\Gamma) = \operatorname{card} (\Gamma \cap \mathbb{Z}^2)$ the number of its integral points.\\

In 1927, J\'{a}rnik proved (indeed in a slightly more general setting) that $N(\Gamma) = O(\ell(\Gamma)^{2/3})$, showing this result to be best possible and providing the optimal value for the constant implied in the $O$ symbol. In 1988, Georges Grekos considered \emph{flat} curves, i.e. curves for which $\ell$ is small compared to $r = r(\Gamma)$, the minimal value of the radius of curvature of $\Gamma$; he proved the upper bound $N(\Gamma) = O(\ell(\Gamma)/r(\Gamma)^{1/3})$; moreover he showed that this result is best possible for $\log(\ell)/\log(r) \in [2/3, 1]$, and he even showed that this is also the case when one fixes the tangent at one extremity of $\Gamma$.\\

The case when $\alpha = \alpha(\Gamma) = \log(\ell(\Gamma))/\log(r(\Gamma)) \in [1/3, 2/3]$ remained unclear for some time. Grekos and I recently noticed that when the tangent at one extremity of $\Gamma$ is parallel to the vector $(1, 0)^t$ then one has $N = O(\ell^2/r)$, which is $o(\ell/r^{1/3})$ when $\alpha < 2/3$, but however, that for any $\alpha \in [1/3, 2/3]$ there exist curves for which $N \gg \ell/r^{1/3}$.\\

I'll present the history of this problem and a recent joint contribution withAdrian Ubis Martinez, giving, for $\alpha \in [1/3, 2/3]$ the best possible upper bound for $N$ in terms of $\ell, r$ and diophantine properties of the tangent to $\Gamma$ at one of its extremity.