I discussed this question with Udi Hrushovski a few months ago.

If I am not mistaken, one can use constructive Nullstellensatz to obtain a similar result. In fact, Theorem 1 of this paper: Sharp estimates for the arithmetic Nullstellensatz (T. Krick, L.M. Pardo, M. Sombra)

seems to give better bounds of the form |A| << log(log(p)) to some power depending on s, where p is the characteristic of the finite field. ]]>

*[Corrected, thanks – T.]*

*[Corrected, thanks – T.]*

(For the other conditions to be possible, even if trivial, we need s>=1 and |A|>=0, in which case we have 0 2^(2^(2^1)) = 16.)

]]>