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

TL;DR

This paper establishes inverse theorems relating sumset sizes and $L^q$ norms of convolutions in discretized $\,\mathbb{R}^d$, with applications to fractal dimensions and uncertainty principles.

Contribution

It extends inverse theorems for sumsets and convolutions to higher dimensions using entropy structure results, broadening previous one-dimensional findings.

Findings

Inverse theorems for sumsets in $\,\mathbb{R}^d$

Bounds on $L^q$ norms of convolutions in discretized spaces

Applications to fractal dimensions and uncertainty principles

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.