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The sum-product estimate for large subsets of prime fields

Author(s): M. Z. Garaev
Journal: Proc. Amer. Math. Soc. 136 (2008), 2735-2739.
MSC (2000): Primary 11B75, 11T23
Posted: April 14, 2008
MathSciNet review: 2399035
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Abstract | References | Similar articles | Additional information

Abstract: Let $\mathbb{F}_p$ be the field of prime order It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $\vert A\vert= N$ such that

$\displaystyle \max\{\vert A+A\vert, \vert AA\vert\}\ll p^{1/2}\vert A\vert^{1/2}.$

One of the results of the present paper implies that if $A\subset \mathbb{F}_p$ with $\vert A\vert>p^{2/3},$ then

$\displaystyle \max\{\vert A+A\vert, \vert AA\vert\}\gg p^{1/2}\vert A\vert^{1/2}.$

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4.: 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), 380-398. MR 2225493 (2007e:11092)
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Additional Information:

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
Email: garaev@matmor.unam.mx

DOI: 10.1090/S0002-9939-08-09386-6
PII: S 0002-9939(08)09386-6
Keywords: Sum-product estimates, prime field, number of solutions.
Received by editor(s): June 26, 2007
Posted: April 14, 2008
Communicated by: Ken Ono
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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