New sum-product type estimates over finite fields

Oliver Roche-Newton; Misha Rudnev; Ilya D. Shkredov

doi:10.1016/j.aim.2016.02.019

New sum-product type estimates over finite fields

Oliver Roche-Newton, Misha Rudnev, Ilya D. Shkredov

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Abstract

Abstract Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F. They are derived from the theorem that the number of incidences between m points and n planes in the projective three-space P G ( 3 , F ) , with m ≥ n = O ( p 2 ) , is O ( m n + k m ) , where k denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that(1) | A ± A | + | A ⋅ A | = Ω ( | A | 1 + 1 5 ) , for any A such that | A | < p 5 8 .

Original language	English
Pages (from-to)	589-605
Number of pages	17
Journal	Advances in Mathematics
Volume	293
Early online date	1 Mar 2016
DOIs	https://doi.org/10.1016/j.aim.2016.02.019
Publication status	Published - 30 Apr 2016

Bibliographical note

This is a revised version: Theorem 1 was incorrect as stated. We give its correct statement; this does not seriously affect the main arguments throughout the paper. Also added is a seres of remarks, placing the result in the context of the current state of the art

Keywords

Sum-product estimates

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Professor Misha Rudnev
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title = "New sum-product type estimates over finite fields",

abstract = "Abstract Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F. They are derived from the theorem that the number of incidences between m points and n planes in the projective three-space P G ( 3 , F ) , with m ≥ n = O ( p 2 ) , is O ( m n + k m ) , where k denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that(1) | A ± A | + | A ⋅ A | = Ω ( | A | 1 + 1 5 ) , for any A such that | A | < p 5 8 .",

keywords = "Sum-product estimates",

author = "Oliver Roche-Newton and Misha Rudnev and Shkredov, {Ilya D.}",

note = "This is a revised version: Theorem 1 was incorrect as stated. We give its correct statement; this does not seriously affect the main arguments throughout the paper. Also added is a seres of remarks, placing the result in the context of the current state of the art",

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T1 - New sum-product type estimates over finite fields

AU - Roche-Newton, Oliver

AU - Rudnev, Misha

AU - Shkredov, Ilya D.

N1 - This is a revised version: Theorem 1 was incorrect as stated. We give its correct statement; this does not seriously affect the main arguments throughout the paper. Also added is a seres of remarks, placing the result in the context of the current state of the art

PY - 2016/4/30

Y1 - 2016/4/30

N2 - Abstract Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F. They are derived from the theorem that the number of incidences between m points and n planes in the projective three-space P G ( 3 , F ) , with m ≥ n = O ( p 2 ) , is O ( m n + k m ) , where k denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that(1) | A ± A | + | A ⋅ A | = Ω ( | A | 1 + 1 5 ) , for any A such that | A | < p 5 8 .

AB - Abstract Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F. They are derived from the theorem that the number of incidences between m points and n planes in the projective three-space P G ( 3 , F ) , with m ≥ n = O ( p 2 ) , is O ( m n + k m ) , where k denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that(1) | A ± A | + | A ⋅ A | = Ω ( | A | 1 + 1 5 ) , for any A such that | A | < p 5 8 .

KW - Sum-product estimates

U2 - 10.1016/j.aim.2016.02.019

DO - 10.1016/j.aim.2016.02.019

M3 - Article (Academic Journal)

SN - 0001-8708

VL - 293

SP - 589

EP - 605

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

New sum-product type estimates over finite fields

Abstract

Bibliographical note

Keywords

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