A classification of Reifenberg properties

TL;DR

This paper classifies twelve variants of Reifenberg's affine approximation properties, exploring their regularity and measure-theoretic characteristics, and constructs counterexamples with connections to number theory to demonstrate the necessity of complexity.

Contribution

It introduces a comprehensive classification of Reifenberg properties and constructs counterexamples linking geometric regularity with number theory, highlighting the necessity of complexity in these properties.

Findings

Different Reifenberg properties correspond to varying regularity levels.

Counterexamples are quasi-self-similar sets connected to number theory.

Complexity in counterexamples is shown to be necessary.

Abstract

We define twelve variants of a Reifenberg's affine approximation property, which are known to be connected with the singular sets of minimal surfaces. With this motivation we investigate the regularity of the sets possessing these. We classify the properties with respect to whether $j$-dimensional Hausdorff dimension, locally finite $j$-dimensional Hausdorff measure or countable $j$-rectifiability hold. In showing that varying levels of regularity hold for the differing properties, quasi-self-similar sets, interesting in their own right, are constructed as counter examples. These counter examples also admit a connection to number theory via the use of the normal number theorem. Additionally, the intriguing result that such complexity in the counter examples is actually a necessity is shown.