Skip to main content
SpringerLink
Log in

Integral points close to a space curve

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We establish sharp lower and upper bounds for the number of integral points near dilations of a space curve with nowhere vanishing torsion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The result there is more precise than the one quoted here.

References

  1. Beresnevich, V.: Rational points near manifolds and metric Diophantine approximation. Ann. Math. (2) 175(1), 187–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beresnevich, V., Dickinson, D., Velani, S.: Diophantine approximation on planar curves and the distribution of rational points With an Appendix II by R. C. Vaughan. Ann. Math. (2) 166(2), 367–426 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beresnevich, V., Vaughan, R.C., Velani, S., Zorin, E.: Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory. Int. Math. Res. Not. IMRN 10, 2885–2908 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Beresnevich, V., Vaughan, R.C., Velani, S., Zorin, E.: Diophantine approximation on curves and the distribution of rational points: divergence theory. arXiv:1809.06159 (arXiv preprint)

  5. Beresnevich, V., Zorin, E.: Explicit bounds for rational points near planar curves and metric Diophantine approximation. Adv. Math. 225(6), 3064–3087 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bombieri, E., Pila, J.: The number of integral points on arcs and ovals. Duke Math. J. 59, 337–357 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ellenberg, J., Venkatesh, A.: On uniform bounds for rational points on non-rational curves. Int. Math. Res. Not. 35, 2163–2181 (2005)

    Article  MATH  Google Scholar 

  8. Harman, G.: Metric Number Theory. London Mathematical Society Monographs. New Series, 18. Oxford University Press, New York (1998)

    Google Scholar 

  9. Heath-Brown, D.R.: The density of rational points on curves and surfaces. Ann. Math. 155, 553–595 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, J.-J.: Rational points near planar curves and Diophantine approximation. Adv. Math. 274, 490–515 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, J.-J.: Hausdorff theory of dual approximation on planar curves. J. Reine Angew. Math. 740, 63–76 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, J.-J.: The density of rational points near hypersurfaces. arXiv:1711.01390 (arXiv preprint)

  13. Huang, J.-J.: Diophantine approximation on the parabola with non-monotonic approximation functions. To appear in Math. Proc. Camb. Philos. Soc. arXiv:1802.00525 (arXiv preprint)

  14. Huang, J.-J., Liu, J.: Simultaneous approximation on affine subspaces. arXiv:1811.06531 (arXiv preprint)

  15. Huang, J.-J., Li, H.: On two lattice points problems about the parabola. arXiv:1902.06047 (arXiv preprint)

  16. Huxley, M.N.: The rational points close to a curve. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21(3), 357–375 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Huxley, M.N.: Area, Lattice Points, and Exponential Sums. London Mathematical Society Monographs. New Series, 13. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1996)

    Google Scholar 

  18. Jarník, V.: Über die Gitterpunkte auf konvexen Kurven. Math. Z. 24(1), 500–518 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  19. Letendre, P.: Topics in analytic number theory. Ph.D. thesis, Université Laval (2018)

  20. Lettington, M.C.: Integer points close to convex surfaces. Acta Arith. 138(1), 1–23 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lettington, M.C.: Integer points close to convex hypersurfaces. Acta Arith. 141(1), 73–101 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Montgomery, H.: Ten lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics (Book 84), American Mathematical Society (1994)

  23. Pila, J.: Geometric postulation of a smooth function and the number of rational points. Duke Math. J. 63(2), 449–463 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schmidt, W.M.: Integer points on curves and surfaces. Monatsh. Math. 99(1), 45–72 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Simmons, D.: Some manifolds of Khinchin type for convergence. J. Théor. Nombres Bordeaux 30(1), 175–193 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sprindžuk, V.G.: Metric theory of diophantine approximations. Wiley, Oxford (1979)

    Google Scholar 

  27. Swinnerton-Dyer, H.P.F.: The number of lattice points on a convex curve. J. Number Theory 6, 128–135 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. Trifonov, S.: Lattice points close to a smooth curve and squarefull numbers in short intervals. J. Lond. Math. Soc. (2) 65(2), 303–319 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vaughan, R.C., Velani, S.: Diophantine approximation on planar curves: the convergence theory. Invent. Math. 166(1), 103–124 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Walsh, M.: Bounded rational points on curves. Int. Math. Res. Not. IMRN 14, 5644–5658 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the anonymous referee for carefully reading the manuscript and providing helpful suggestions to improve the presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Nevada, Reno, 1664 N. Virginia St., Reno, NV, 89557, USA

    Jing-Jing Huang

Authors
  1. Jing-Jing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing-Jing Huang.

Additional information

Communicated by Kannan Soundararajan.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research is supported by the UNR VPRI startup Grant 1201-121-2479.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, JJ. Integral points close to a space curve. Math. Ann. 374, 1987–2003 (2019). https://doi.org/10.1007/s00208-019-01832-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-019-01832-5

Mathematics Subject Classification

Search

Navigation