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

 

Fourier analysis and expanding phenomena in finite fields


Authors: Derrick Hart, Liangpan Li and Chun-Yen Shen
Journal: Proc. Amer. Math. Soc. 141 (2013), 461-473
MSC (2010): Primary 11B75
Published electronically: June 19, 2012
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Abstract: In this paper we study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by other authors using spectral graph theory. In addition, several generalizations of these results are given.

In the case that $ A$ is a subset of a prime field $ \mathbb{F}_p$ of size less than $ p^{1/2}$ it is shown that $ \vert\{a^2+b:a,b \in A\}\vert\geq C_1 \vert A\vert^{147/146}$ and $ \vert\{\frac {b+1}{a}:a,b \in A\}\vert\geq C_2 \vert A\vert^{110/109}$, where $ \vert\cdot \vert$ denotes the cardinality of a set and $ C_1$ and $ C_2$ are absolute constants.


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Additional Information

Derrick Hart
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email: dnhart@math.rutgers.edu

Liangpan Li
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China – and – Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, United Kingdom
Email: liliangpan@gmail.com

Chun-Yen Shen
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
Address at time of publication: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: shenc@umail.iu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11338-3
PII: S 0002-9939(2012)11338-3
Keywords: Sums, products, expanding maps, character sums
Received by editor(s): April 10, 2011
Received by editor(s) in revised form: July 4, 2011
Published electronically: June 19, 2012
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.