Integral Points on a Very Flat Convex Curve

Abstract

The second named author studied in 1988 the possible relations between the length $\ell$, the minimal radius of curvature r and the number of integral points N of a strictly convex flat curve in $\mathbb {R}^2$, stating that $N = O(\ell /r^{1/3})$ (*), a best possible bound even when imposing the tangent at one extremity of the curve; here flat means that one has $\ell = r^{\alpha }$ for some $\alpha \in [2/3, 1)$. He also proved that when $\alpha \le 1/3$, the quantity N is bounded. In this paper, the authors prove that in general the bound (*) cannot be improved for very flat curves, i.e. those for which $\alpha \in (1/3, 2/3)$; however, if one imposes a 0 tangent at one extremity of the curve, then (*) is replaced by the sharper inequality $N \le \ell ^2/r +1$.

To Krishnaswami Alladi, for his 60th birthday

Jean-Marc Deshouillers acknowledges the support of the binational research project MuDeRa, funded by the French and Austrian Science Funds ANR and FWF. The research of Georges Grekos has been supported by the French grant CAESAR ANR-12-BS01-0011.