There is a paper by Bourgain, Katz and Tao

*Bourgain, Jean; Katz, N.; Tao, Terence C.*, **A sum-product estimate in finite fields, and applications**, Geom. Funct. Anal. 14, No. 1, 27-57 (2004). ZBL1145.11306.

where they proved:

$$ \begin{array}{l}{\text { Theorem } 1.1 \text { (Sum-product estimate). } \text {Let } F :=\mathbf{Z} / q \mathbf{Z} \text { for some prime } q} \\ {\text { and let } A \text { be a subset of } F \text { such that }} \\ {\qquad|F|^{\delta}<|A|<|F|^{1-\delta}} \\ {\text { for some } \delta>0 \text { . Then one has a bound of the form }} \\ {\max (|A+A|,|A \cdot A|) \geq c(\delta)|A|^{1+\varepsilon}} \\ {\text { for some } \varepsilon=\varepsilon(\delta)>0}\end{array} $$

What is conjectured to be the value of $\epsilon$?