Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T09:17:30.481Z Has data issue: false hasContentIssue false
  • English
  • Français

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

Published online by Cambridge University Press:  01 January 2009

J. BOURGAIN  and
M. Z. GARAEV
J. BOURGAIN
Affiliation:
Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, U.S.A. e-mail: bourgain@math.ias.edu
M. Z. GARAEV
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3 (Xangari), C.P. 58089, Morelia, Michoacán, México. e-mail: garaev@matmor.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than . This implies that if p1/2 < M < p, then

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 1 - 21
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bourgain, J.Estimates on exponential sums related to the Diffie–Hellman distributions. Geom. Funct. Anal. 15 (2005), 1–34.CrossRefGoogle Scholar
[2]Bourgain, J.Mordell's exponential sum estimate revisited. J. Amer. Math. Soc. 18 (2005), 477–499.CrossRefGoogle Scholar
[3]Bourgain, J.More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (2005), 1–32.CrossRefGoogle Scholar
[4]Bourgain, J. The sum-product theorem in Z q with q arbitrary. Preprint.Google Scholar
[5]Bourgain, J. Multilinear exponential sum bounds with optimal entropy assignments. Geom. Funct. Anal. (to appear).Google Scholar
[6]Bourgain, J. and Chang, M.-C.Exponential sum estimates over subgroups and almost subgroups of , where Q is composite with few prime factors. Geom. Funct. Anal. 16 (2006), 327–366.CrossRefGoogle Scholar
[7]Bourgain, J., Glibichuk, A. A. and Konyagin, S. V.Estimates for the number of sums and products and for exponential sums in fields of prime order. J. London Math. Soc. (2) 73 (2006), 380–398.CrossRefGoogle Scholar
[8]Bourgain, J., Katz, N. and Tao, T.A sum-product estimate in finite fields and their applications, Geom. Func. Anal. 14 (2004), 27–57.CrossRefGoogle Scholar
[9]Bourgain, J. and Konyagin, S. V.Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. C. R. Acad. Sci. Paris, Ser I 337 (2003), 75–80.CrossRefGoogle Scholar
[10]Garaev, M. Z.An explicit sum-product estimate in F p. Int. Math. Res. Notices, Vol. 2007 (2007), doi:10.1093/imrn/rnm035.Google Scholar
[11]Glibichuk, A. A. and Konyagin, S. V. Additive properties of product sets in fields of prime order. Centre de Recherches Mathématiques, CRM Proceedings and Lecture Notes, 43, 279–286 (2007).CrossRefGoogle Scholar
[12]Katz, N. H. and Shen, Ch.-Y. A slight improvement to Garaev's sum product estimate. Proc. Amer. Math. Soc. (to appear).Google Scholar
[13]Konyagin, S. V. Bounds of exponential sums over subgroups and Gauss sums. Proc. 4th Intern. Conf. Modern Problems of Number Theory and Its Applications, Moscow Lomonosov State Univ., Moscow (2002), 86–114 (in Russian).Google Scholar
[14]Ruzsa, I. Z.An application of graph theory to additive number theory. Scientia, Ser. A 3 (1989), 97–109.Google Scholar
[15]Ruzsa, I. Z.Sums of finite sets. Number theory (New York, 1991–1995), 281–293 (Springer, 1996).Google Scholar
[16]Tao, T. and Vu, V.Additive Combinatorics (Cambridge University Press, 2006).CrossRefGoogle Scholar
[17]Vinogradov, I. M.An Introduction to the Theory of Numbers (Pergamon Press, 1955).Google Scholar