Sum-product phenomenon in finite fields not of prime order
type: Item
instance of: scholarly article
title: Sum-product phenomenon in finite fields not of prime order
zbMATH Open document ID: 1270.11019
DOI: 10.1216/RMJ-2011-41-3-941
author: Chun-Yen Shen
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-08-02T00:00:00Z
review text: Let \(F=\mathbb F_{p^n}\) be a finite field, and let \(A\) be a subset of \(F\) so that for any \(A'\subset A\) with \(|A'|\geq |A|^{15/16}\) and for any subfield \(G\subset F\) and for any elements \(c,d\in F\) if \[ A'\subset cG+d, \] then \[ |A'|\leq |G|^{1/2}. \] Then it must be that \[ \max(|A+A|, |f(A,A)|) \gtrapprox |A|^{17/16} \] where \(f: \mathbb F_p\times \mathbb F_p \rightarrow \mathbb F_p\) is a function defined by \(f(x,y)=x(g(x)+cy)\), where \(c\in \mathbb F_p^*\) and \(g: \mathbb F_p \rightarrow \mathbb F_p\) is any function.
reviewed by: Weidong Gao
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 5934877
zbMATH Keywords: sum-product estimates
zbMATH Keywords: expanding maps
MaRDI profile type: Publication
cites work: ESTIMATES FOR THE NUMBER OF SUMS AND PRODUCTS AND FOR EXPONENTIAL SUMS IN FIELDS OF PRIME ORDER
cites work: A sum-product estimate in finite fields, and applications
cites work: An Explicit Sum-Product Estimate in Fp
cites work: Sum-product Estimates in Finite Fields via Kloosterman Sums
cites work: A slight improvement to Garaev's sum product estimate
cites work: Q3400049
cites work: An extension of Bourgain and Garaev's sum-product estimates
cites work: The sum-product phenomenon in arbitrary rings
cites work: Q5393666
description: scientific article; zbMATH DE number 5934877