  In this paper we consider estimating the number of solutions to
multiplicative equations in finite fields when the variables run through
certain sets with high additive structure. In particular, we consider
estimating the multiplicative energy of generalized arithmetic progressions in
prime fields and of boxes in arbitrary finite fields and obtain sharp bounds in
more general scenarios than previously known. Our arguments extend some ideas
of Konyagin and Bourgain and Chang into new settings.
