Israel Journal of Mathematics

, Volume 215, Issue 1, pp 503–512 | Cite as

A quantitative Oppenheim theorem for generic diagonal quadratic forms

  • Jean BourgainEmail author


We establish a quantitative version of Oppenheim’s conjecture for one-parameter families of ternary indefinite quadratic forms using an analytic number-theory approach. The statements come with power gains and in some cases are essentially optimal.


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  1. [Bl]
    V. Blomer, Epstein zeta-functions, subconvexity, and the purity conjecture, arXiv:1602.02326.Google Scholar
  2. [B-B-R-R]
    V. Blomer, J. Bourgain, M. Radziwill and Z. Rudnick, Small gaps in the spectrum of the rectangular billiard, arXiv:160402413c2.Google Scholar
  3. [Bo]
    J. Bourgain, On pair correlation for generic diagonal forms, preprint 2016 arXiv:1606.06173.Google Scholar
  4. [E-M-M]
    A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Annals of mathematics 147 (1998), 93–141.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [G-K]
    A. Ghosh and D. Kelmer, Shrinking targets for semisimple groups, arXiv:151205848.Google Scholar
  6. [Ju]
    M. Jutila, Zero-density estimates for L-functions, Acta Arithmetica 32 (1977), 55–62.MathSciNetzbMATHGoogle Scholar
  7. [L-M]
    E. Lindenstrauss and G. Margulis, Effective estimates on indefinite ternary forms, Israel Journal of Mathematics 203 (2014), 445–499.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [M]
    G. Margulis, Formes quadratiques indéfinies et flots unipotents sur les espaces homog`enes, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 304 (1987), 249–253.MathSciNetzbMATHGoogle Scholar
  9. [S]
    P. Sarnak, Values at integers of binary quadratic forms, in Harmonic Analysis and Number Theory (Montreal, PQ, 1996), CMS Conference Proceedings, Vol. 21, American Mathematical Society, Providence, RI, 1997, pp. 181–203.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA

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