New sum-product type estimates over finite fields

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

Research output: Contribution to journalArticle (Academic Journal)peer-review

43 Citations (Scopus)
278 Downloads (Pure)

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 languageEnglish
Pages (from-to)589-605
Number of pages17
JournalAdvances in Mathematics
Volume293
Early online date1 Mar 2016
DOIs
Publication statusPublished - 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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