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A Second Wave of Expanders in Finite Fields

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

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 than 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.

Keywords

  • Sum product problem
  • Incidence bounds
  • Collinear triples
  • Arithmetic combinatorics

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Acknowledgements

We thank Olly Roche-Newton, Misha Rudnev, and Sophie Stevens for several helpful suggestions. We would also like to thank Mel Nathanson for inviting us to write this survey for the proceedings of CANT 2015/2016.

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Murphy, B., Petridis, G. (2017). A Second Wave of Expanders in Finite Fields. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_15

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