  We prove inverse theorems for the size of sumsets and the $L^q$ norms of
convolutions in the discretized setting, extending to arbitrary dimension an
earlier result of the author in the line. These results have applications to
the dimensions of dynamical self-similar sets and measures, and to the higher
dimensional fractal uncertainty principle. The proofs are based on a structure
theorem for the entropy of convolution powers due to M.~Hochman.
