On the Distance Sets of Self-Similar Sets

TL;DR

This paper proves that self-similar sets in the plane with positive length have distance sets of Hausdorff dimension one, revealing a fundamental geometric property of such fractal sets.

Contribution

It establishes a new result linking positive length of self-similar sets to the Hausdorff dimension of their distance sets.

Findings

Distance set of positive-length self-similar sets in the plane has Hausdorff dimension one.

Provides a significant geometric property of self-similar fractals.

Advances understanding of the structure of fractal distance sets.

Abstract

We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.