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Minkowski dimension for measures

Kenneth J. Falconer, Jonathan M. Fraser, Antti Käenmäki

Abstract

The purpose of this article is to introduce and motivate the notion of Minkowski (or box) dimension for measures. The definition is simple and fills a gap in the existing literature on the dimension theory of measures. As the terminology suggests, we show that it can be used to characterise the Minkowski dimension of a compact metric space. We also study its relationship with other concepts in dimension theory.

Minkowski dimension for measures

Abstract

The purpose of this article is to introduce and motivate the notion of Minkowski (or box) dimension for measures. The definition is simple and fills a gap in the existing literature on the dimension theory of measures. As the terminology suggests, we show that it can be used to characterise the Minkowski dimension of a compact metric space. We also study its relationship with other concepts in dimension theory.

Paper Structure

This paper contains 5 sections, 9 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Minkowski dimension
  3. Packing dimension
  4. Assouad spectrum and $L^q$-dimensions
  5. Lower spectrum

Key Result

Theorem 2.1

If $X$ is a compact metric space, then

Figures (1)

  • Figure 1: Illustration for the set $X$ in Example \ref{['ex:lower-strict']}.

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Example 3.2
  • Lemma 3.3
  • proof
  • Example 3.4
  • Theorem 3.5
  • proof
  • ...and 11 more