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A note on higher integrability of projections

Tuomas Orponen

TL;DR

The paper answers a sharp question about higher $L^{p}$-integrability of projections of $t$-Frostman measures. It introduces a δ-discretised framework to produce a $(\delta,t)$-set with controlled projection structures via a stick-based construction, then stitches these discretised pieces into a Cantor-type limit measure $\mu$ supported on a fractal set. The main achievement is constructing $\mu$ with $\pi_{\theta}\mu \notin L^{p}$ for every $\theta$ and all $p>2/(2-t)$, showing the integrability threshold is optimal in this uniform sense. This resolves a question of Peres and Schlag and sharpens the understanding of projection behavior for fractal measures, while leaving open refined mixed-norm and exceptional-projection questions for future work.

Abstract

Let $t \in [1,2)$ and $p > 2/(2 - t)$. I construct a $t$-Frostman Borel measure $μ$ on $[0,1]^{2}$ such that $π_θμ\notin L^{p}$ for every $θ\in S^{1}$. This answers a question of Peres and Schlag.

A note on higher integrability of projections

TL;DR

The paper answers a sharp question about higher -integrability of projections of -Frostman measures. It introduces a δ-discretised framework to produce a -set with controlled projection structures via a stick-based construction, then stitches these discretised pieces into a Cantor-type limit measure supported on a fractal set. The main achievement is constructing with for every and all , showing the integrability threshold is optimal in this uniform sense. This resolves a question of Peres and Schlag and sharpens the understanding of projection behavior for fractal measures, while leaving open refined mixed-norm and exceptional-projection questions for future work.

Abstract

Let and . I construct a -Frostman Borel measure on such that for every . This answers a question of Peres and Schlag.

Paper Structure

This paper contains 9 sections, 3 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work and open problems
  3. A $\delta$-discretised counterpart of Theorem \ref{['thm:main']}
  4. Projections of a single stick
  5. Arrangement of sticks: preliminaries
  6. Case $t \in [1,\tfrac{4}{3})$
  7. Case $t \in [\tfrac{4}{3},2)$
  8. The $(\delta,t)$-set property
  9. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $t \in [1,2)$ and $p > 2/(2 - t)$. Then, there exists a $t$-Frostman Borel measure $\mu$ on $[0,1]^{2}$ such that $\pi_{\theta}\mu \notin L^{p}(\mathbb{R})$ for every $\theta \in S^{1}$.

Figures (3)

  • Figure 1: Depiction of $\mathcal{P}$ for $t \in [\tfrac{4}{3},2)$. The red sticks have length $\delta^{1 - t/2}$. As the parameter $t$ increases, the number of "cartwheels" in the picture decreases, and the separation of the sticks decreases. In the extreme case $t = 2$, there would be only one cartwheel of diameter $\sim 1$.
  • Figure 2: Depiction of $\mathcal{P}$ for $t \in [1,4/3)$. The red sticks have length $\delta^{1 - t/2}$.
  • Figure 3: Since $B(x,r)$ does not intersect $B(c_{j},\delta^{1 - t/2})$, the "radial projection" $J$ of $B(x,3r)$ to $\partial B_{j}$ has length $|J| \sim r$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Definition 2.1: $(\delta,t,C)$-set
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • proof : Proof of Proposition \ref{['mainProp']}
  • Claim 2.5
  • proof
  • Claim 2.6
  • Definition 3.1: Cantor construction
  • ...and 3 more