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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fourier analysis and expanding phenomena in finite fields
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by Derrick Hart, Liangpan Li and Chun-Yen Shen PDF
Proc. Amer. Math. Soc. 141 (2013), 461-473 Request permission

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 $|\{a^2+b:a,b \in A\}|\geq C_1 |A|^{147/146}$ and $|\{\frac {b+1}{a}:a,b \in A\}|\geq C_2 |A|^{110/109}$, where $|\cdot |$ 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
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 461-473
  • MSC (2010): Primary 11B75
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11338-3
  • MathSciNet review: 2996950