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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The sum-product estimate for large subsets of prime fields


Author: M. Z. Garaev
Journal: Proc. Amer. Math. Soc. 136 (2008), 2735-2739
MSC (2000): Primary 11B75, 11T23
Published electronically: April 14, 2008
MathSciNet review: 2399035
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Abstract: Let $ \mathbb{F}_p$ be the field of prime order $ p.$ 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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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: http://dx.doi.org/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
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.