Garaev's inequality in Finite Fields not of prime order

In the present paper, we extend Garaev’s techniques to the set of fields which are not necessarily of prime order. Our goal here is just to find an explicit estimate in the supercritical setting where the set A has less cardinality than the square root of the cardinality of the field, and interacts in a less than half-dimensional way with any subfields. (We make this precise below.) Precisely, we