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 thatAAis a subset of a prime fieldFp\mathbb F_pof size less thanp1/2p^{1/2}it is shown that|{a2+b:a,b∈A}|≥C1|A|147/146|\{a^2+b:a,b \in A\}|\geq C_1 |A|^{147/146}and|{b+1a:a,b∈A}|≥C2|A|110/109|\{\frac {b+1}{a}:a,b \in A\}|\geq C_2 |A|^{110/109}, where|⋅||\cdot |denotes the cardinality of a set andC1C_1andC2C_2are absolute constants.