Who Asked Us? How the Theory of Computing Answers Questions about Analysis

TL;DR

This paper surveys how algorithmic fractal dimensions, rooted in computability theory, have been applied to resolve open questions in classical geometric measure theory and mathematical analysis.

Contribution

It reviews recent applications of computability-based fractal dimensions to classical analysis problems and discusses future research directions.

Findings

Algorithmic fractal dimensions have successfully addressed open problems in geometric measure theory.

Recent developments connect computability theory with classical mathematical analysis.

The survey highlights promising future research avenues in this interdisciplinary area.

Abstract

Algorithmic fractal dimensions -- constructs of computability theory -- have recently been used to answer open questions in classical geometric measure theory, questions of mathematical analysis whose statements do not involve computability theory or logic. We survey these developments and the prospects for future such results.