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A restricted projection problem for fractal sets in $\mathbb{R}^n$

Shengwen Gan, Shaoming Guo, Hong Wang

Abstract

Let $γ: [-1, 1]\to \mathbb{R}^n$ be a smooth curve that is non-degenerate. Take $m\le n$ and a Borel set $E\subset [0, 1]^n$. We prove that the orthogonal projection of $E$ to the $m$-th order tangent space of $γ$ at $θ\in [-1, 1]$ has Hausdorff dimension $\min\{m, \dim(E)\}$ for almost every $θ\in [-1, 1]$.

A restricted projection problem for fractal sets in $\mathbb{R}^n$

Abstract

Let be a smooth curve that is non-degenerate. Take and a Borel set . We prove that the orthogonal projection of to the -th order tangent space of at has Hausdorff dimension for almost every .
Paper Structure (5 sections, 2 theorems, 133 equations)

This paper contains 5 sections, 2 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Idea of the proof
  3. Frequency decomposition and decoupling inequalities: Step One.
  4. Frequency decomposition and decoupling inequalities: General steps
  5. Output of the algorithm

Key Result

Theorem 1.2

Let $n\ge 2, 1\le m\le n$ and $\bm{\gamma}: [-1, 1]\to \mathbb{R}^n$ be a non-degenerate curve. Let $E\subset [0, 1]^n$ be a Borel measurable set. Then for almost every $\theta\in [-1, 1]$. Here $\textup{dim}$ refers to the Hausdorff dimension.

Theorems & Definitions (8)

  • Conjecture 1.1: Fässler-Orponen
  • Theorem 1.2
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['221013theorem1_1']} by assuming Theorem \ref{['221013theorem2_1']}.
  • Claim 3.1
  • proof : Proof of Claim \ref{['221027claim3_1']}
  • Claim 4.1
  • proof : Proof of Claim \ref{['221026claim4_1']}