Title: A Second Wave of Expanders over Finite Fields,
Abstract: This is an expository survey on recent sum-product results in finite fields.
We present a number of sum-product or "expander" results that say that if
$|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$
is nearly as large as possible, and if $|A|<p^{2/3}$ then the set in question
is significantly larger that $A$. These results are based on a point-plane
incidence bound of Rudnev, and are quantitatively stronger than a wave of
earlier results following Bourgain, Katz, and Tao's breakthrough sum-product
result.
In addition, we present two geometric results: an incidence bound due to
Stevens and de Zeeuw, and bound on collinear triples, and an example of an
expander that breaks the threshold of $p^{2/3}$ required by the other results.
We have simplified proofs wherever possible, and hope that this survey may
serve as a compact guide to recent advances in arithmetic combinatorics over
finite fields. We do not claim originality for any of the results.