New sum-product type estimates over finite fields

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• 作者：Oliver Roche-Newton^a ; ^{o.rochenewton@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Misha Rudnev^b ; ^{m.rudnev@bristol.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Ilya D. Shkredov^c ; ^d ; ^{ilya.shkredov@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：68R05 ; 11B75
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：30 April 2016
• 年：2016
• 卷：293
• 期：Complete
• 页码：589-605
• 全文大小：403 K

文摘

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 PG(3,F)$PG(3,F)$, with m≥n=O(p²)$m\ge n=O(p^2)$, is 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

for any A   such that $|A|<p^{\frac{5}{8}}$.