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A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS

Part of: Designs and configurations Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  24 May 2019

CHANGHAO CHEN ,
BRYCE KERR  and
ALI MOHAMMADI
CHANGHAO CHEN
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
BRYCE KERR*
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
ALI MOHAMMADI
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia email [email protected]
*
✉[email protected]
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Abstract

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We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if $A\subseteq \mathbb{F}_{p}$ satisfies $|A|\leq p^{64/117}$ then $\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$ Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018, arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^{+}(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^{+}(P)$ to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.

Keywords

sum–product estimateadditive combinatorics

MSC classification

Primary: 11T99: None of the above, but in this section
Secondary: 05B10: Difference sets (number-theoretic, group-theoretic, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 268 - 280
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The first and second author were supported by ARC Grant DP170100786.

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