Geodesic
On the Number of Rational Points on a Strictly Convex Curve
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 30-42

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Let $\gamma$ be a bounded convex curve on the plane. Then $\#(\gamma\cap(\mathbb{Z}/n)^2)=o(n^{2/3})$. This strengthens the classical result due to Jarník (the upper bound $cn^{2/3}$) and disproves the conjecture on the existence of a so-called universal Jarník curve.
Keywords: convex curve, lattice point, affine length. 
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     title = {On the {Number} of {Rational} {Points} on a {Strictly} {Convex} {Curve}},
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F. V. Petrov. On the Number of Rational Points on a Strictly Convex Curve. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 30-42. http://geodesic.mathdoc.fr/item/FAA_2006_40_1_a2/