An explicit sum-product estimate in $\mathbb{F}_p$

Details An explicit sum-product estimate in $\mathbb{F}_p$ An explicit sum-product estimate in $\mathbb{F}_p$

2007-02-26

arxiv.org/abs/math/0702780v1

Mathematics

Number Theory

11 pages

Scientific paper

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a non-empty subset of $\mathbb{F}_p.$ In this paper we give an explicit version of the sum-product estimate of Bourgain, Katz, Tao and Bourgain, Glibichuk, Konyagin on the size of $\max\{|A+A|, |AA|\}.$ In particular, our result implies that if $1<|A|\le p^{7/13}(\log p)^{-4/13},$ then $$ \max\{|A+A|, |AA|\}\gg \frac{|A|^{15/14}}{(\log|A|)^{2/7}} . $$

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