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On the projections of almost Ahlfors regular sets

Tuomas Orponen, Kevin Ren

Abstract

We show that the "sharp Kaufman projection theorem" from 2023 is sharp in the class of Ahlfors $(1,δ^{-ε})$-regular sets. This is in contrast with a recent result of the first author, which improves the projection theorem in the class of Ahlfors $(1,C)$-regular sets.

On the projections of almost Ahlfors regular sets

Abstract

We show that the "sharp Kaufman projection theorem" from 2023 is sharp in the class of Ahlfors -regular sets. This is in contrast with a recent result of the first author, which improves the projection theorem in the class of Ahlfors -regular sets.

Paper Structure

This paper contains 4 sections, 9 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main construction
  4. Proof of Theorem \ref{['main2']}

Key Result

Theorem 1.3

For every $C > 0$ and $0 < t < \tau \leq 1$, there exist $\delta_{0},\epsilon > 0$ such that the following holds for all $\delta \in (0,\delta_{0}]$. Let $\mu$ be an Ahlfors $(1,\delta^{-\epsilon})$-regular probability measure with $K := \operatorname{spt} \mu \subset B(1)$, and let $\nu$ be a Borel

Figures (2)

  • Figure 1: The case $10\rho^{j} \leq r \leq \rho^{j - 1} \cdot \rho^{(1 + \tau)/2}$.
  • Figure 2: Case $\rho^{j} \cdot \rho^{(1 + \tau)/2} \leq r \leq 10\rho^{j - 1}$.

Theorems & Definitions (18)

  • Definition 1.1: Ahlfors $(s,C)$-regularity
  • Theorem 1.3: Corollary 4.9 in 2023arXiv230110199O
  • Theorem 1.5
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.11
  • Definition 2.1: $(s,C)$-regularity between two scales
  • Proposition 2.3
  • proof
  • ...and 8 more