Inverse theorems for discretized sums and Lq norms of convolutions in Rd

Abstract

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.