Skip to main content
SpringerLink
Log in

Lattice point problems and values of quadratic forms

  • Published:
Inventiones mathematicae Aims and scope

Abstract

For d-dimensional ellipsoids E with d≥5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order \(\mathcal{O}(r^{d-2})\) for general ellipsoids and up to an error of order o(r d-2) for irrational ones. The estimate refines earlier bounds of the same order for dimensions d≥9. As an application a conjecture of Davenport and Lewis about the shrinking of gaps between large consecutive values of Q[m],m∈ℤd of positive definite irrational quadratic forms Q of dimension d≥5 is proved. Finally, we provide explicit bounds for errors in terms of certain Minkowski minima of convex bodies related to these quadratic forms.

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

References

  1. Bentkus, V., Götze, F.: On the lattice point problem for ellipsoids. Acta Arith. LXXX.2, 101–125 (1997)

    Google Scholar 

  2. Bentkus, V., Götze, F.: Lattice Point Problems and Distribution of Values of Quadratic Forms. Ann. Math. 150, 977–1027 (1999)

    Article  MathSciNet  Google Scholar 

  3. Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin, Göttingen, Heidelberg: Springer 1959

  4. Cook, R.J., Raghavan, S.: Indefinite quadratic polynomials of small signature. Monatsh. Math. 97, 169–176 (1984)

    Article  MathSciNet  Google Scholar 

  5. Davenport, H.: Indefinite quadratic forms in many variables (II). Proc. Lond. Math. Soc. 8, 109–126 (1958)

    Article  MathSciNet  Google Scholar 

  6. Davenport, H., Lewis, D.J.: Gaps between values of positive definite quadratic forms, Acta Arith. 22, 87–105 (1972)

    Google Scholar 

  7. Eskin, A., Margulis, G.A., Mozes, S.: Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. Math. 147, 93–141 (1998)

    Article  MathSciNet  Google Scholar 

  8. Fricker, F.: Einführung in die Gitterpunktlehre. Basel, Boston, Stuttgart: Birkhäuser 1982

  9. Götze, F., Gordin, M.: Limiting distributions of theta series’ on Siegel half spaces. St. Petersburg Math. J. 15, 81–102 (2004)

    Article  MathSciNet  Google Scholar 

  10. Hardy, G.: On Dirichlet’s divisor problem. Proc. Lond. Math. Soc. 15, 1–25 (1917)

    Article  Google Scholar 

  11. Hlawka, E.: Über Integrale auf konvexen Körpern I, II. Monatsh. Math. 54, 1–36, 81–99 (1950)

    Article  MathSciNet  Google Scholar 

  12. Jarnik, V.: Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Ann. 100, 699–721 (1928)

    Article  MathSciNet  Google Scholar 

  13. Jarnik, V., Walfisz, A.: Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Z. 32, 152–160 (1930)

    Article  MathSciNet  Google Scholar 

  14. Jurkat, W.B., Van Horne, J.W.: On the central limit theorem for theta series. Mich. Math. J. 29, 65–77 (1982)

    Article  MathSciNet  Google Scholar 

  15. Krätzel, E., Nowak, G.: Lattice points in large convex bodies. Monatsh. Math. 112, 61–72 (1991)

    Article  MathSciNet  Google Scholar 

  16. Krätzel, E., Nowak, G.: Lattice points in large convex bodies, II. Acta Arith. LXII.3, 285–295 (1992)

  17. Landau, E.: Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Z. 21, 126–132 (1924)

    Article  MathSciNet  Google Scholar 

  18. Landau, E., Walfisz, A.: Ausgewählte Abhandlungen zur Gitterpunktlehre, ed. by A. Walfisz. Berlin: VEB Deutscher Verlag der Wissenschaften 1962

  19. Lewis, D.J.: The distribution of the values of real quadratic forms at integer points. In: Analytic number theory (Proc. Sympos. Pure Math., XXIV, St. Louis Univ. (1972)), pp. 159–174. Providence 1973

  20. Margulis, G.A.: In: Discrete subgroups and ergodic theory, Number theory, trace formulas and discrete groups (Oslo, 1987), pp. 377–398. Boston: Academic Press 1989

  21. Margulis, G. A.: Oppenheim conjecture. Fields Medallists’ lectures, pp. 272–327. River Edge: World Scientific 1997

  22. Marklof, J.: Limit theorems for theta sums. Duke Math. J. 97, 127–153 (1999)

    Article  MathSciNet  Google Scholar 

  23. Matthes, T.K.: The multivariate Central Limit Theorem for regular convex sets. Ann. Prob. 3, 503–515 (1970)

    Article  MathSciNet  Google Scholar 

  24. Mumford, D.: Tata Lectures on Theta I. Boston, Basel, Stuttgart: Birkhäuser 1983

  25. Szegö, G.: Beiträge zur Theorie der Laguerreschen Polynome II. Zahlentheoretische Anwendungen. Math. Z. 25, 388–404 (1926)

    Google Scholar 

  26. Walfisz, A.: Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Z. 19, 300–307 (1924)

    Article  MathSciNet  Google Scholar 

  27. Walfisz, A.: Teilerprobleme. Math. Z. 26, 66–88 (1927)

    Article  MathSciNet  Google Scholar 

  28. Walfisz, A.: Gitterpunkte in mehrdimensionalen Kugeln. Warszawa: Państwowe Wydawnictwo Naukowe 1957

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Friedrich Götze

Authors
  1. Friedrich Götze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Friedrich Götze.

Additional information

Mathematics Subject Classification (1991)

11P21

Rights and permissions

Reprints and permissions

About this article

Cite this article

Götze, F. Lattice point problems and values of quadratic forms. Invent. math. 157, 195–226 (2004). https://doi.org/10.1007/s00222-004-0366-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-004-0366-3

Keywords

Search

Navigation