Dimensions and metric dyadic cubes

TL;DR

This paper demonstrates that various fractal dimensions, including Hausdorff, Minkowski, and Assouad, can be equivalently characterized using dyadic cube systems in metric spaces with finite Assouad dimension.

Contribution

It establishes the equivalence of multiple fractal dimensions with dyadic cube constructions in doubling metric spaces, extending the applicability of dyadic methods.

Findings

Dimensions can be expressed via dyadic cubes in doubling spaces.

The Assouad spectrum is also compatible with dyadic cube characterizations.

Dyadic cube systems provide a unified framework for various fractal dimensions.

Abstract

In this note, we provide equivalent definitions for fractal geometric dimensions through dyadic cube constructions. Given a metric space $X$ with finite Assouad dimension, i.e., satisfying the doubling property, we show that the construction of systems of dyadic cubes by Hyt\"onen-Kairema is compatible with many dimensions. In particular, the Hausdorff, Minkowski, and Assouad dimensions can be equivalently expressed solely using dyadic cubes in the aforementioned system. The same is true for the Assouad spectrum, a collection of dimensions introduced by Fraser-Yu.