## 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

This is a preview of subscription content, access via your institution.

## Buying options

## References

M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, M. Rudnev, Group actions and geometric combinatorics in \({\mathbb{F}_q^d}\). 11 2013

J. Bourgain, M.Z. Garaev, On a variant of sum-product estimates and explicit exponential sum bounds in prime fields. Math. Proc. Camb. Philos. Soc.

**146**(1), 1–21 (2009)J. Bourgain, N. Katz, T. Tao, A sum-product estimate in finite fields, and applications. Geom. Funct. Anal.

**14**(1), 27–57 (2004)J. Bourgain, Multilinear exponential sums in prime fields under optimal entropy condition on the sources. Geom. Funct. Anal.

**18**(5), 1477–1502 (2009)J. Chapman, M. Burak Erdogan, D. Hart, A. Iosevich, D. Koh,

*Pinned Distance Sets, k-simplices, Wolff’s Exponent in Finite Fields and Sum-Product Estimates*(2009)F. de Zeeuw, A short proof of Rudnev’s point-plane incidence bound (2016). arXiv:1612.02719

G. Elekes, On the number of sums and products. Acta Arith.

**81**(4), 365–367 (1997)P. Erdos, E. Szemerédi, On sums and products of integers. Stud. Pure Math. 213–218 (1983)

M.Z. Garaev, An explicit sum-product estimate in \({\mathbb{F}_p}\). Int. Math. Res. Not. IMRN, (11):Art. ID rnm035, 11 (2007)

M.Z. Garaev, The sum-product estimate for large subsets of prime fields. Proc. Am. Math. Soc.

**136**(8), 2735–2739 (2008)A.A. Glibichuk, S.V. Konyagin, Additive properties of product sets in fields of prime order, in

*Additive Combinatorics*. CRM Proceedings and Lecture Notes, vol. 43 (2007), pp. 279–286D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture. Trans. Am. Math. Soc.

**363**(6), 3255–3275 (2011)D. Hart, A. Iosevich, J. Solymosi, Sum-product estimates in finite fields via Kloosterman sums. Int. Math. Res. Not. IMRN, (5):Art. ID rnm007, 14 (2007)

N.H. Katz, C.-Y. Shen, A slight improvement to Garaev’s sum product estimate. Proc. Am. Math. Soc.

**136**(7), 2499–2504 (2008)B. Lund, S. Saraf,

*Incidence Bounds for Block Designs*(2014)B. Murphy, G. Petridis, A point-line incidence identity in finite fields, and applications. Mosc. J. Comb. Number Theory

**6**(1), 64–95 (2016)B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev, I.D. Shkredov, New results on sum-product type growth in positive characteristic (2017)

B. Murphy, O. Roche-Newton, I. Shkredov, Variations on the sum-product problem. SIAM J. Discrete Math.

**29**(1), 514–540 (2015)G. Petridis, Collinear triples and quadruples for Cartesian products in \({\mathbb{F}_p^2}\) (2016)

G. Petridis, Pinned algebraic distances determined by Cartesian products in \({\mathbb{F}_p^2}\) (2016)

G. Petridis, Products of differences in prime order finite fields (2016)

G. Petridis, I.E. Shparlinski, Bounds on trilinear and quadrilinear exponential sums (2016)

T. Pham, L.A. Vinh, F. de Zeeuw, Three-variable expanding polynomials and higher-dimensional distinct distances (2016). arXiv:1612.09032

O. Roche-Newton, A short proof of a near-optimal cardinality estimate for the product of a sum set (2015)

O. Roche-Newton, M. Rudnev, I.D. Shkredov, New sum-product type estimates over finite fields. Adv. Math.

**293**, 589–605 (2016)M. Rudnev, An improved sum-product inequality in fields of prime order. Int. Math. Res. Not. IMRN

**16**, 3693–3705 (2012)M. Rudnev, On the number of incidences between planes and points in three dimensions. Combinatorica (2014) (To appear)

M. Rudnev, I.D. Shkredov, S. Stevens, On the energy variant of the sum-product conjecture (2016)

I.D. Shkredov, On a question of A. Balog. Pacific J. Math.

**280**(1), 227–240 (2016)S. Stevens, F. de Zeeuw, An improved point-line incidence bound over arbitrary fields (2016)

L.A. Vinh, The Szemerédi-trotter type theorem and the sum-product estimate in finite fields. Eur. J. Combin.

**32**(8), 1177–1181 (2011)E.A. Yazici, B. Murphy, M. Rudnev, I. Shkredov, Growth estimates in positive characteristic via collisions. Int. Math. Res. Not, IMRN (2016)

## 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2017 Springer International Publishing AG

## About this paper

### Cite this paper

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

### Download citation

DOI: https://doi.org/10.1007/978-3-319-68032-3_15

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-68030-9

Online ISBN: 978-3-319-68032-3

eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)