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 \(N \le \ell ^2/r +1\).