Fourier analysis and expanding phenomena in finite fields

Derrick Hart, Liangpan Li, Chun-Yen Shen

Published 2009-09-30Version 1

In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu 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 $|\{a^2+b:a,b \in A\}|\geq C |A|^{147/146}$, where $|\cdot|$ denotes the cardinality of the set and $C$ is an absolute constant.