Skip to main content
SpringerLink
Log in

Exponential sum estimates over a subgroup in an arbitrary finite field

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Let F q be the finite field consisting of q = p r elements and yy an additive character of the field F q . Take an arbitrary multiplicative subgroup H of size |H| > q C/(log log q) for some constant C > 0 not largely contained in any multiplicative shift of a subfield. We show that |Σ h∈H yy(h)| = o(|H|). This means that H is equidistributed in F q .

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. J. Bourgain, Multilinear exponential sums in prime fields under optimal entropy condition on the sources, Geom. Funct. Anal. 18 (2009), 1477–1502.

    Article  MathSciNet  Google Scholar 

  2. J. Bourgain, On exponential sums in finite fields, preprint.

  3. J. Bourgain and M. Z. Garaev, On a variant of sum-product estimates and explicit exponential sums bounds in prime fields, Math. Proc. Cambridge Philos. Soc. 146 (2009), 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Z. Garaev, A quantified version of Bourgain’s sum-product estimate in F p for subsets of incomparable sizes, Electron. J. Combin. 15 (2008), Research paper 58.

    Google Scholar 

  5. M. Z. Garaev, Sums and products of sets and estimates for rational trigonometric sums in fields of prime order, Uspekhi Mat. Nauk, 65 (2010), 5–66, translation in Russian Math. Surveys 65 (2010), 599–658.

    MathSciNet  Google Scholar 

  6. N. Katz and C. Shen, Garaev’s inequality in finite fields not of prime order, Online J. Anal. Comb. No. 3 (2008), Art. 3.

  7. I. Z. Ruzsa, An application of graph theory to additive number theory, Sci. Ser. A Math. Sci. (N.S.) 3 (1989), 97–109.

    MathSciNet  MATH  Google Scholar 

  8. I. Z. Ruzsa, Sums of finite sets, in Number Theory New York, 1991–1995, Springer, New York, 1996, pp. 281–293.

    Chapter  Google Scholar 

  9. B. Sudakov, E. Szemeredi and V. Vu, On a question of Erdös and Moser, Duke Math. J., 129 (2005), 129–155.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, Cambridge, 2006.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Olden Lane, Princeton, NJ, 08540, USA

    J. Bourgain

  2. Insituto de Matemáticas, Universidad Nacional Autónoma de México Campusmorelia, Apardado Potal 61-3 (Xangari), C. P. 580899, Moreliamicaocán, México

    A. Glibichuk

Authors
  1. J. Bourgain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Glibichuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Bourgain.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bourgain, J., Glibichuk, A. Exponential sum estimates over a subgroup in an arbitrary finite field. JAMA 115, 51–70 (2011). https://doi.org/10.1007/s11854-011-0023-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-011-0023-x

Keywords

Search

Navigation