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On the Hewitt Stromberg dimension of product sets

Najmeddine Attia

Abstract

In this paper, we construct new multifractal measures, on the Euclidean space $\mathbb{R}^n$, in a similar manner to Hewitt-Stomberg measures but using the class of all $n$-dimensional half-open binary cubes of covering sets in the definition rather than the class of all balls. As an application we shall be concerned with evaluation of Hewitt-Stromberg dimension of cartesian product sets by means of the dimensions of their components.

On the Hewitt Stromberg dimension of product sets

Abstract

In this paper, we construct new multifractal measures, on the Euclidean space , in a similar manner to Hewitt-Stomberg measures but using the class of all -dimensional half-open binary cubes of covering sets in the definition rather than the class of all balls. As an application we shall be concerned with evaluation of Hewitt-Stromberg dimension of cartesian product sets by means of the dimensions of their components.

Paper Structure

This paper contains 6 sections, 13 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Relation between ${\mathsf H}^t$ and $\overline{{\mathsf H}}^t$
  4. Construction of the multifractal measures
  5. Application : Cartesian products of sets
  6. Example

Key Result

Lemma 1

Let $B$ is a ball in $\mathbb R^n$ of diameter $\delta >0$. The number of balls of diameter $\gamma \in (0, \delta)$ necessary to cover $B$ is less then

Theorems & Definitions (26)

  • Remark 1
  • Definition 1
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 16 more