Integral points on strictly convex closed curves

  • S. V. Konyagin


A negative answer is given to Swinnerton-Dyer's question: Is it true that for any ε > 0 there exists a positive integer n such that for any planar closed strictly convex n-times differentiable curve Γ, when it is blown up a sufficiently large number ν of times, the number of integral points on the resultant curve will be less than νɛ. An example has been constructed when this number for an infinite number ν is not less than ν1/2, while Γ is infinitely differentiable.


Positive Integer Infinite Number Integral Point Negative Answer Closed Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    W. Jarnik, “Über die Gitterpunkte auf convexen Kurven,” Math. Z.,24, 500–518 (1925).Google Scholar
  2. 2.
    E. Landau, Vorlesungen über Zahlentheorie, Bd. 2, Hirzel, Leipzig (1927).Google Scholar
  3. 3.
    H. P. F. Swinnerton-Dyer, “The number of lattice points on a convex curve,” J. Number Theory,6, 128–135 (1974).Google Scholar
  4. 4.
    V. S. Panferov, “Analogs of the Luzin-Denjoy and Cantor-Lebesgue theorems for double trigonometric series,” Mat. Zametki,18, No. 5, 659–674 (1975).Google Scholar
  5. 5.
    Loo-keng Hua, “On the number of solutions of Tarry's problem,” Acta Sci. Sinica,1, No. 1, 1–76 (1953).Google Scholar
  6. 6.
    V. S. Panferov, “Properties of convergent and absolutely convergent multiple trigonometric Series,” Candidate's Dissertation, Moscow (1976).Google Scholar

Copyright information

© Plenum Publishing Corporation 1977

Authors and Affiliations

  • S. V. Konyagin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityUSSR

Personalised recommendations