Inverse theorems for discretized sums and $L^q$ norms of convolutions in ${\mathbb{R}}^d$

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.

19 pages. v3: many corrections and clarifications, incorporates referees' comments, main results unchanged

References in corpus (4)

• The structure theory of set addition revisited
• An introduction to fractal uncertainty principle
• New bounds on the dimensions of planar distance sets
• A higher dimensional Bourgain-Dyatlov fractal uncertainty principle