The sum-product estimate for large subsets of prime fields

Garaev, M.

doi:10.1090/S0002-9939-08-09386-6

The sum-product estimate for large subsets of prime fields
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by M. Z. Garaev PDF

Proc. Amer. Math. Soc. 136 (2008), 2735-2739 Request permission

Abstract:

Let $\mathbb {F}_p$ be the field of 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 \[ \max \{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. \] One of the results of the present paper implies 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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