The sum-product estimate for large subsets of prime fields

Details The sum-product estimate for large subsets of prime fields The sum-product estimate for large subsets of prime fields

2007-06-05

arxiv.org/abs/0706.0702v1

Mathematics

Number Theory

7 pages

Scientific paper

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for
any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with
$|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the
present paper we prove that if $A\subset \mathbb{F}_p$ with $|A|>p^{2/3},$ then
$$ \max\{|A+A|, |AA|\}\gg p^{1/2}|A|^{1/2}. $$

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