Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Pablo Shmerkin

TL;DR

This work develops high-dimensional inverse theorems for discretized sumsets and $L^q$ norms of convolutions in $\mathbb{R}^d$ by leveraging Hochman’s entropy-saturation framework. The authors show that, under near-optimal convolution behavior, one can extract large, scale-wise uniform subsets whose structure is controlled by a sequence of subspaces $(V_{sL})$ and almost-saturation on $k_s$-dimensional planes, with a complementary subset contained in $k_s$-planes. The results connect to the dimension theory of dynamical self-similar measures, additive energy reduction, and the Fractal Uncertainty Principle, and they integrate uniformization lemmas, Balog–Szemerédi–Gowers arguments, and entropy methods into a robust, higher-dimensional inverse theory. These contributions provide both conceptual insight into how non-smoothing at scale manifests as near-linear-algebraic structure and practical tools for applications in fractal geometry and harmonic analysis.

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.

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

TL;DR

This work develops high-dimensional inverse theorems for discretized sumsets and norms of convolutions in by leveraging Hochman’s entropy-saturation framework. The authors show that, under near-optimal convolution behavior, one can extract large, scale-wise uniform subsets whose structure is controlled by a sequence of subspaces and almost-saturation on -dimensional planes, with a complementary subset contained in -planes. The results connect to the dimension theory of dynamical self-similar measures, additive energy reduction, and the Fractal Uncertainty Principle, and they integrate uniformization lemmas, Balog–Szemerédi–Gowers arguments, and entropy methods into a robust, higher-dimensional inverse theory. These contributions provide both conceptual insight into how non-smoothing at scale manifests as near-linear-algebraic structure and practical tools for applications in fractal geometry and harmonic analysis.

Abstract

We prove inverse theorems for the size of sumsets and the 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.
Paper Structure (11 sections, 14 theorems, 92 equations)

This paper contains 11 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction and main results
  2. Uniformization lemmas
  3. Proof of Theorem \ref{['thm:inverse-thm-A+A']}
  4. Hochman's theorem on saturation of self-convolutions
  5. Proof of Theorem \ref{['thm:inverse-thm-A+A']}
  6. Proof of Theorem \ref{['thm:inverse']}
  7. Connections and applications
  8. Dimensions of dynamical self-similar measures, and applications
  9. Khalil's inverse theorem
  10. Additive energy of discretized sets
  11. Application to the Fractal Uncertainty Principle

Key Result

Theorem 1.2

For each $q\in (1,\infty)$ and $\delta>0$ the following holds if $L\ge L(\delta)\in\mathbb{N}$ and $\varepsilon<\varepsilon(L)$, for all sufficiently large $S=S(\delta,L,\varepsilon)$: Let $m=SL$. Suppose $\mu,\nu$ are $2^{-m}$-measures on $[0,1)^d$ such that Then there exist $2^{-m}$-sets $A\subset \mathop{\mathrm{supp}}\nolimits\mu$ and $B\subset\mathop{\mathrm{supp}}\nolimits\nu$ and a sequenc

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 16 more