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A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS

  • CHANGHAO CHEN (a1), BRYCE KERR (a2) and ALI MOHAMMADI (a3)

Abstract

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.

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

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[1] Bourgain, J. and Garaev, M. Z., ‘On a variant of sum-product estimates and explicit exponential sum bounds in prime fields’, Math. Proc. Cambridge Philos. Soc. 146(1) (2009), 121.
[2] Bourgain, J., Katz, N. and Tao, T., ‘A sum-product estimate in finite fields and their applications’, Geom. Funct. Anal. 14 (2004), 2757.
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[4] Erdős, P. and Szemerédi, E., ‘On sums and products of integers’, in: Studies in Pure Mathematics. To the memory of Paul Turán (Birkhäuser, Basel, 1983), 213218.
[5] Garaev, M. Z., ‘An explicit sum-product estimate in 𝔽p ’, Int. Math. Res. Not. IMRN 2007 (2007), Article ID 11, 11 pages.
[6] Garaev, M. Z., ‘The sum-product estimate for large subsets of prime fields’, Proc. Amer. Math. Soc. 136 (2008), 27352739.
[7] Glibichuk, A. A. and Konyagin, S. V., ‘Additive properties of product sets in fields of prime order’, in: Additive Combinatorics, CRM Proceedings and Lecture Notes, 43 (American Mathematical Society, Providence, RI, 2007), 279286.
[8] Katz, N. H. and Shen, C. Y., ‘A slight improvement to Garaev’s sum product estimate’, Proc. Amer. Math. Soc. 136 (2008), 24992504.
[9] Konyagin, S. V. and Rudnev, M., ‘On new sum-product type estimates’, SIAM J. Discrete Math. 27(2) (2013), 973990.
[10] Li, L., ‘Slightly improved sum-product estimates in fields of prime order’, Acta Arith. 147 (2011), 153160.
[11] Murphy, B., Roche-Newton, O. and Shkredov, I., ‘Variations of the sum-product problem’, SIAM J. Discrete Math. 29(1) (2015), 514540.
[12] Murphy, B., Petridis, G., Roche-Newton, O., Rudnev, M. and Shkredov, I. D., ‘New results on sum-product type growth over fields’, Preprint, 2017, arXiv:1702.01003.
[13] Murphy, B., Rudnev, M., Shkredov, I. and Shteinikov, Y., ‘On the few products, many sums problem’, Preprint, 2017, arXiv:1712.0041v1.
[14] Roche-Newton, O., Rudnev, M. and Shkredov, I. D., ‘New sum-product type estimates over finite fields’, Adv. Math. 293 (2016), 589605.
[15] Rudnev, M., ‘An improved sum-product inequality in fields of prime order’, Int. Math. Res. Not. IMRN 2012(16) (2012), 36933705.
[16] Rudnev, M., ‘On the number of incidences between points and planes in three dimensions’, Combinatorica 38(1) (2017), 219254.
[17] Rudnev, M., Shakan, G. and Shkredov, I., ‘Stronger sum-product inequalities for small sets’, Preprint, 2018, arXiv:1808.08465.
[18] Shakan, G., ‘On higher energy decomposition and the sum-product phenomenon’, Math. Proc. Came. Phil. Soc., to appear.
[19] Shakan, G. and Shkredov, I. D., ‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018, arXiv:1806.07091v1.
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[21] Shkredov, I. D., ‘On asymptotic formulae in some sum-product questions’, Preprint, 2018, arXiv:1802.09066.
[22] Stevens, S. and de Zeeuw, F., ‘An improved point-line incidence bound over arbitrary fields’, Bull. Lond. Math. Soc. 49(5) (2017), 842858.
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A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS

  • CHANGHAO CHEN (a1), BRYCE KERR (a2) and ALI MOHAMMADI (a3)

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