Algorithmic Fractal Dimensions in Geometric Measure Theory

TL;DR

This paper surveys the development of algorithmic fractal dimensions and their applications in geometric measure theory, highlighting connections with computable functions and recent theorem-proving in classical fractal geometry.

Contribution

It provides a comprehensive overview of how algorithmic dimensions have advanced understanding in fractal geometry and suggests future research directions.

Findings

Algorithmic fractal dimensions connect computability with geometric measure theory.

Recent applications include proving new theorems in classical fractal geometry.

The survey identifies promising future research directions.

Abstract

The development of algorithmic fractal dimensions in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. We survey these developments, with emphasis on connections with computable functions on the reals, recent uses of algorithmic dimensions in proving new theorems in classical (non-algorithmic) fractal geometry, and directions for future research.