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Ekström-Persson conjecture regarding random covering sets

Esa Järvenpää, Maarit Järvenpää, Markus Myllyoja, Örjan Stenflo

Abstract

We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii $(n^{-α})_{n=1}^\infty$. For generating balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.

Ekström-Persson conjecture regarding random covering sets

Abstract

We consider the Hausdorff dimension of random covering sets generated by balls and general measures in Euclidean spaces. We prove, for a certain parameter range, a conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii . For generating balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.
Paper Structure (7 sections, 14 theorems, 128 equations, 1 figure)

This paper contains 7 sections, 14 theorems, 128 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Results
  3. Preliminary lemmas
  4. Proof of Theorem \ref{['theorem:generallowerboundenergy']}
  5. Proof of Theorem \ref{['theorem:nminusalpha']}
  6. Proof of Theorem \ref{['theorem:lowerboundforgeneralballs']}
  7. Sharpness of the bounds in Theorem \ref{['theorem:lowerboundforgeneralballs']}

Key Result

Theorem 2.2

Let $\mu\in \mathcal{P}(\mathbb R^d)$ and $\alpha>0$. If $\frac{1}{\alpha}<\mathop{\mathrm{\overline{\mathrm{dim}}_{\mathrm{H}}}}\nolimits\mu$, then where

Figures (1)

  • Figure 1: The almost sure dimension $f_{\mu}(\underline{r})$ depicted as a "function" of $s_2(\underline{r})$.

Theorems & Definitions (31)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Remark 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Lemma 3.1
  • proof
  • ...and 21 more