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arXiv · 2308.09846

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

Pablo Shmerkin
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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.

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Pablo Shmerkin. 2023-08-18. Inverse theorems for discretized sums and $L^q$ norms of convolutions in $\mathbb{R}^d$. https://doi.org/10.2140/om.2025.2.65

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