The sum-product estimate for large subsets of prime fields

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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Cited by in corpus (1)

• The sum-product phenomenon in arbitrary rings