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 \(k(\gamma, n) = o(n^{2/3})\).

\[ k(\gamma(N),N) := \#(\gamma(N) \cap L_N)\geq cN^{2/3} + o(N^{2/3}).\tag{*} \]

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 \(k(\gamma, n) = o(n^{2/3})\).

Reviewer: Olaf Ninnemann (Berlin)

##### 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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\textit{F. V. Petrov}, Funct. Anal. Appl. 40, No. 1, 24--33 (2006; Zbl 1152.11030); translation from Funkts. Anal. Prilozh. 40, No. 1, 30--42 (2006)

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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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