Fourier analysis and expanding phenomena in finite fields

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.