# The sum-product estimate for large subsets of prime fields - Mathematics > Number Theory

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Abstract: Let $\mathbb{F} p$ be the field of a prime order $p.$ It is known that forany 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 thepresent paper we prove that if $A\subset \mathbb{F} p$ with $|A|>p^{2-3},$ then$$\max\{|A+A|, |AA|\}\gg p^{1-2}|A|^{1-2}.$$

Autor: M. Z. Garaev

Fuente: https://arxiv.org/