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On the projections of Ahlfors regular sets in the plane

Tuomas Orponen

Abstract

This paper contains the following $δ$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,ε> 0$ and $s \in [0,1]$, there exists $κ> 0$ such that the following holds for all $δ> 0$ small enough. Let $ν$ be a Borel probability measure on $S^{1}$ satisfying $ν(B(x,r)) \leq Cr^ε$ for all $x \in S^{1}$ and $r > 0$. Let $K \subset B(1) \subset \mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $θ\in \mathrm{spt\,} ν$ such that $$|π_θ(F)|_δ \geq δ^{ε- s}$$ for all $F \subset K$ with $|F|_δ \geq δ^{κ- s}$. Here $π_θ(z) = θ\cdot z$ for $z \in \mathbb{R}^{2}$.

On the projections of Ahlfors regular sets in the plane

Abstract

This paper contains the following -discretised projection theorem for Ahlfors regular sets in the plane. For all and , there exists such that the following holds for all small enough. Let be a Borel probability measure on satisfying for all and . Let be Ahlfors -regular with constant at most . Then, there exists a vector such that for all with . Here for .

Paper Structure

This paper contains 25 sections, 29 theorems, 426 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof sketch and paper structure
  3. Technical lemmas
  4. Lemmas on high multiplicity sets
  5. Entropy
  6. A Lebesgue differentiation lemma on $[0,1]$
  7. Finding branching scales for Frostman measures
  8. The main proposition
  9. Proof of Proposition \ref{['mainTechnicalProp']}
  10. Finding a tangent with exact dimensional projections
  11. Proof of Proposition \ref{['mainTechnicalProp']}
  12. Defining the constant $\gamma$ and the set $\mathcal{Q} \subset [0,1]$
  13. Proof of the main proposition
  14. Recap and applying Proposition \ref{['mainTechnicalProp']}
  15. Two good directions
  16. ...and 10 more sections

Key Result

Theorem 1.3

For every $C > 0$, $s,\tau \in (0,1]$, and $\underline{s} \in [0,s)$, there exist $\kappa = \kappa(C,s,\underline{s},\tau) > 0$ and $\delta_{0} = \delta_{0}(C,s,\underline{s},\tau) > 0$ such that the following holds for all $\delta \in (0,\delta_{0}]$. Let $\nu$ be a Borel probability measure on $S^ for all Borel sets $F \subset B(1)$ with $\mu(F) \geq \delta^{\kappa}$.

Figures (1)

  • Figure 1: The squares $Q_{1}',\ldots,Q_{p}' \in \mathcal{D}_{j + 1}$ intersecting $\eta_{Q}$, the points $x_{1}',\ldots,x_{p}' \subset F \cap Q \cap T$ and the points $y_{1}',\ldots,y_{p}' \subset F_{50\Delta^{a_{j + 1}}} \cap \pi_{\theta}^{-1}\{\pi_{\theta}(x)\}$. The larger ball $B(x,50\Delta^{a_{j}})$ has not been drawn in the figure, but it is large enough to contain $Q \in \mathcal{D}_{j}$, and especially all the points $y_{1}',\ldots,y_{p}'$.

Theorems & Definitions (73)

  • Definition 1.2: Ahlfors $(s,C)$-regularity
  • Theorem 1.3
  • Corollary 1.5
  • proof
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9: Upper $(s,C)$-regularity
  • Definition 1.10: $(s,C)$-Frostman measure
  • Definition 1.11: $(s,C)$-regularity
  • ...and 63 more