Integral points on convex curves
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Integral points on convex curves Jean-Marc Deshouillers1 · Adrián Ubis2 To the memory of Javier Cilleruelo. Received: 16 November 2018 / Accepted: 22 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We estimate the maximal number of integral points which can be on a convex arc in R2 with given length, minimal radius of curvature and initial slope. Keywords Lattice points · Convex curves · Farey fractions Mathematics Subject Classification 11P21
1 Introduction Evaluating the number of integral points (points with integral coordinates) on finite continuous curves in R2 is a fairly general Diophantine question. Since the distance between two distinct elements in Z2 is at least 1, on a simple curve with length there cannot be more than + 1 integral points, a bound which is only achieved for some linear curves. Besides the study of specific curves, the first general result is due to Jarník [5] who proved in 1925 that the number of points on a strictly convex arc y = f (x) of length is at most 3(4π )−1/3 2/3 + O(1/3 ), and that this bound is reached for some arc.
This work has been initiated during a visit of J-M D. at ICMAT (Madrid); both authors have been partially supported by the Grant MTM2014-56350-P. J-M D. also acknowledges the support on the CEFIPRA Project 5401 and the ANR-FWF Project MuDeRa.
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Adrián Ubis [email protected] Jean-Marc Deshouillers [email protected]
1
Institut de Mathématiques de Bordeaux, Université de Bordeaux, Bordeaux INP et CNRS, 33405 Talence, France
2
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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J. Deshouillers, A. Ubis
From there Jarník deduced a similar result for strictly convex simple closed curves, giving the optimal bound 3(2π )−1/3 2/3 + O(1/3 ). In 1963, Andrews [1] gave an upper bound for the number N of integral points on the boundary of a strictly convex body in Rn in terms of the volume V of that body, which is N V 1/3 when n = 2. Grekos [4], in 1988, revisited Jarník’s method in the case of strictly convex flat C 2 curves, i.e. curves for which the length = () is smaller than the minimum of the radius of curvature along . Denoting by r = r () this minimal radius of curvature and by N = N () the number of integral points on , he first obtains the upper bound N ≤ 2r −1/3 .
(1.1)
With an unspecified constant, this result can be derived from [1]. The second result of [4] implies that, up to the constant, (1.1) is the best possible, as long as is not too flat—i.e. log / log r > 2/3—and the lower bound he obtains for families of curves is uniform in terms of the slope w = w() of the curve (i.e. the tangent of its angle with the x-axis). The relevance of the slope is pointed out in [3]: Grekos and the first named author of the present paper showed that for any strictly convex C 2 curve with a tangent at the origin parallel to the x-axis, the number of its integral points satisfies N ≤ 2 /r + /r + 1,
(1.2)
a quantity which is essentia
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