The sumproduct estimate for large subsets of prime fields
Author:
M. Z. Garaev
Journal:
Proc. Amer. Math. Soc. 136 (2008), 27352739
MSC (2000):
Primary 11B75, 11T23
Published electronically:
April 14, 2008
MathSciNet review:
2399035
Fulltext PDF Free Access
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Abstract: Let be the field of prime order It is known that for any integer one can construct a subset with such that One of the results of the present paper implies that if with then
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 J. Bourgain, The sumproduct theorem in with arbitrary, preprint.
 2.
 J. Bourgain, More on the sumproduct phenomenon in prime fields and its applications, Int. J. Number Theory 1 (2005), 132. MR 2172328 (2006g:11041)
 3.
 J. Bourgain and M.C. Chang, Exponential sum estimates over subgroups and almost subgroups of , where is composite with few prime factors, Geom. Funct. Anal. 16 (2006), 327366. MR 2231466 (2007d:11093)
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 J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), 380398. MR 2225493 (2007e:11092)
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 J. Bourgain, N. Katz and T. Tao, A sumproduct estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), 2757. MR 2053599 (2005d:11028)
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 N. H. Katz and Ch.Y. Shen, A slight improvement to Garaev's sum product estimate, preprint.
 11.
 N. H. Katz and Ch.Y. Shen, Garaev's inequality in fields not of prime order, preprint.
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 J. Solymosi, On the number of sums and products, Bull. London Math. Soc. 37 (2005), 491494. MR 2143727 (2006c:11021)
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Additional Information
M. Z. Garaev
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 613 (Xangari), C.P. 58089, Morelia, Michoacán, México
Email:
garaev@matmor.unam.mx
DOI:
http://dx.doi.org/10.1090/S0002993908093866
PII:
S 00029939(08)093866
Keywords:
Sumproduct estimates,
prime field,
number of solutions.
Received by editor(s):
June 26, 2007
Published electronically:
April 14, 2008
Communicated by:
Ken Ono
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
