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arXiv · 0706.0702

The sum-product estimate for large subsets of prime fields

M. Z. Garaev
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Abstract

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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M. Z. Garaev. 2007-06-05. The sum-product estimate for large subsets of prime fields. https://arxiv.org/abs/0706.0702

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