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On the number of rational points on a strictly convex curve. (English. Russian original) Zbl 1152.11030
Funct. Anal. Appl. 40, No. 1, 24-33 (2006); translation from Funkts. Anal. Prilozh. 40, No. 1, 30-42 (2006).
From the author’s introduction: In his famous paper [Math. Z. 24, 500–518 (1926; JFM 51.0153.01)], V. Jarník proved, among other results, that the maximum possible number of integer points that can lie on a strictly convex curve of length $$N$$ on the plane grows as $$cN^{2/3}$$. (The sharp constant $$c$$ was also computed by Jarník and is equal to $$3(2\pi)^{-1/3}$$.) In other words, the number of points of the lattice $$L_N := (\frac 1N \mathbb Z)^2$$ that lie on a strictly convex curve $$\gamma$$ of length 1 does not exceed $$cN^{2/3}$$, and for each $$N$$ there exists a strictly convex curve $$\gamma(N)$$ of length 1 such that
$k(\gamma(N),N) := \#(\gamma(N) \cap L_N)\geq cN^{2/3} + o(N^{2/3}).\tag{*}$
Hence the following natural question arises: does there exist a universal curve $$\gamma$$ for which inequality (*) holds (possibly, with a different constant $$c$$ for infinitely many positive integers $$N$$?
This question was put forward by Vershik and apparently mentioned in the literature for the first time by Plagne [Acta Arith., 87, No. 3, 255–267 (1999; Zbl 0924.11084)], who indicated that he had been introduced to the topic by J.-M. Deshouillers and G. Grekos. H. P. F. Swinnerton-Dyer [J. Number Theory 6, 128–135 (1974; Zbl 0285.10020)] proved that if $$\gamma \in C^3$$, then $$k(\gamma,N)\leq cN^{3/5+\varepsilon}$$. E. Bombieri and J. Pila [Duke Math. J. 59, No. 2, 337–357 (1989; Zbl 0718.11048)] established that $$k(\gamma,N)\leq cN^{1/2+\varepsilon}$$ for each $$\varepsilon > 0$$ provided that $$\gamma$$ is infinitely smooth. We do not mention any further results of this kind related to smoothness, conditions on the curvature, and other restrictions imposed on the curve.
A. M. Vershik [Funct. Anal. Appl. 28, No. 1, 13–20 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 16–25 (1994; Zbl 0848.52004)] and I. Bárány [Discrete Comput. Geom. 13, No. 3-4, 279-295 (1995; Zbl 0824.52001)] discussed the limit shapes of large random polygons with vertices on a fine lattice. The results reveal a connection with affine geometry. Namely, let $$l_a(\gamma)$$ be the affine length of a curve $$\gamma$$ (the integral of the cubic root of the curvature with respect to a natural parameter on the curve). It turns out that the number of polygons with vertices in $$L_n$$ in a small neighborhood of a given curve $$\gamma$$ grows as $$e^{c\cdot l_a(\gamma)n^{2/3}}$$ and the number of their vertices grows as $$c\cdot l_a(\gamma)n^{2/3}$$. (Remarkably, this is true for both the maximal and the typical number of vertices except that the constants are different.) Polygons with vertices in $$L_n$$ contained in a given convex polygon accumulate near a closed convex curve with maximum possible affine length [Bárány, loc. cit.]. This curve is a union of parabola arcs inscribed in the angles of the polygon. Hence the number of nodes of $$L_N$$ on such a curve does not exceed $$C N^{1/2}$$. Thus a typical curve is not universal. Here we prove that in fact there is no universal curve. Namely, the following theorem holds.
Theorem 1. Let $$\gamma$$ be a bounded strictly convex curve. Then $$k(\gamma, n) = o(n^{2/3})$$.

##### MSC:
 11H06 Lattices and convex bodies (number-theoretic aspects) 52C05 Lattices and convex bodies in $$2$$ dimensions (aspects of discrete geometry) 11P21 Lattice points in specified regions
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##### References:
 [1] V. Jarník, ”Über die Gitterpunkte auf konvexen Kurven,” Math. Z., 24, 500–518 (192 · JFM 51.0153.01 [2] A. Plagne, ”A uniform version of Jarn’ik’s theorem,” Acta Arith., 87, No. 3, 255–267 (199 · Zbl 0924.11084 [3] H. P. F. Swinnerton-Dyer, ”The number of lattice points on a convex curve,” J. Number Theory, 6, 128–135 (197 · Zbl 0285.10020 [4] E. Bombieri and J. Pila, ”The number of integral points on arcs and ovals,” Duke Math. J., 59, 337–357 (198 · Zbl 0718.11048 [5] A. M. Vershik, ”The limit shape of convex lattice polygons and related topics,” Funkts. Anal. Prilozhen., 28, No. 1, 16–25 (1994); English transl.: Functional Anal. Appl., 28, No. 1, 13–20 (19 · Zbl 0848.52004 [6] I. Barany, ”The limit shape of convex lattice polygons,” Discrete Comput. Geom., 13, 279–295 (199 [7] J. Favard, Cours de géométrie différentielle locale, Gauthier-Villars, Paris, 1957. · Zbl 0077.15002 [8] A. Eskin and C. McMullen, ”Mixing, counting and equidistribution in Lie groups,” Duke Math. J., 71, 181–209 (199 · Zbl 0798.11025 [9] A. Ya. Khinchin, Continued Fractions [in Russian], Fizmatgiz, Moscow, 1961.
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