The sum-product estimate for large subsets of prime fields

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 $\vert A\vert= N$ such that

$$\max\{\vert A+A\vert, \vert AA\vert\}\ll p^{1/2}\vert A\vert^{1/2}.$$

One of the results of the present paper implies that if $A\subset \mathbb{F}_p$ with $\vert A\vert>p^{2/3},$ then

$$\max\{\vert A+A\vert, \vert AA\vert\}\gg p^{1/2}\vert A\vert^{1/2}.$$