A question on "SUM-PRODUCT...VIA KLOOSTERMANN SUMS", by Hart, Iosevich and Solymosi

Question

In this paper https://arxiv.org/pdf/math/0609426.pdf, the authors, state, as a consequence of Theorem 1.1, the following sum-product estimate.

Theorem 1.1 says that for all $A\subset\mathbb{F}_q$, we have $$|A|^3\ll q^{-1}\cdot |A+A|^2\cdot |A\cdot A|\cdot |A|+q^{1/2}\cdot|A+A|\cdot |A\cdot A|,$$

while the addendum says:

If $A\subset \mathbb{F}_q$ such that $|A|\ll q^{7/10}$, then $$\max(|A+A|,|A\cdot A|)\gg \frac{|A|^{3/2}}{q^{1/4}}$$.

Can someone explain me how they deduce this from Theorem 1.1?

Many thanks!

Answer 1

Let us introduce the notation $$M:=\max(|A+A|,|A\cdot A|).$$ Your first display implies that either $|A|^3\ll q^{1/2}M^2$ or $|A|^3\ll q^{-1}M^3|A|$. In the first case we get $M\gg q^{-1/4}|A|^{3/2}$ without any assumption on $|A|$. In the second case we get $$M\gg q^{1/3}|A|^{2/3}\gg q^{-1/4}|A|^{3/2},$$ where the second inequality follows from $q^{7/10}\gg |A|$. Done.