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arXiv · 1701.01635

A Second Wave of Expanders over Finite Fields

Brendan Murphy·Giorgis Petridis
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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.

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Brendan Murphy, Giorgis Petridis. 2017-01-06. A Second Wave of Expanders over Finite Fields. https://arxiv.org/abs/1701.01635

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