Geodesic
Estimates for the number of rational points on convex curves and surfaces
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XV, Tome 344 (2007), pp. 174-189 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\Gamma\subset \mathbb R^d$ be a bounded strictly convex surface. Denote by $k_n(\Gamma)$ the number of points in the set $\Gamma\cap\frac1n\mathbb Z^d$. We prove that $\liminf k_n(\Gamma)/n^{d-2}<\infty$ for $d\ge 3$ and $\liminf k_n(\Gamma)/\log n<\infty$ for $d=2$. 
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     title = {Estimates for the number of rational points on convex curves and surfaces},
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F. V. Petrov. Estimates for the number of rational points on convex curves and surfaces. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XV, Tome 344 (2007), pp. 174-189. http://geodesic.mathdoc.fr/item/ZNSL_2007_344_a3/

[1] V. Jarník, “Über die Gitterpunkte auf konvexen Kurven”, Math. Z., 24 (1926), 500–518 | DOI | MR

[2] A. Vershik, “Predelnaya forma vypuklykh mnogougolnikov”, Funkts. anal. i ego pril., 28 (1994), 13–20 | MR | Zbl

[3] I. Barany, “The limit shape of convex lattice polygons”, Discrete Comput. Geom., 13 (1995), 279–295 | DOI | MR | Zbl

[4] G. E. Andrews, “A lower bound for the volume of strictly convex bodies with many boundary lattice points”, Trans. Amer. Math. Soc., 106 (1963), 270–279 | DOI | MR | Zbl

[5] E. Werner, “A general geometric construction for affine surface area”, Studia Math., 132:3 (1999), 227–238 | MR | Zbl

[6] P. M. Gruber, “Baire categories in convexity”, Handbook of Convex Geometry, Vol. A, B, North-Holland, Amsterdam, 1993, 1327–1346 | MR | Zbl

[7] I. Barany, “The technique of $M$-regions and cap coverings: a survey”, III International Conference in “Stochastic Geometry, Convex Bodies and Empirical Measures”, Part II (Mazara del Vallo, 1999), 2000, 21–38 | MR | Zbl

[8] I. Barany, A. M. Vershik, “On the number of convex lattice polytopes”, Geom. Funct. Anal., 2:4 (1992), 381–393 | DOI | MR | Zbl

[9] F. V. Petrov, “O kolichestve ratsionalnykh tochek na strogo vypukloi krivoi”, Funkts. anal. i ego pril., 40:1 (2006), 30–42 | MR | Zbl