Self-similar sets and Lipschitz graphs

TL;DR

This paper explores the relationship between self-similar sets and Lipschitz graphs, showing that certain fractal attractors contain large subsets that can be covered by Lipschitz graphs, with explicit dependence on their dimensions.

Contribution

It demonstrates that self-similar attractors satisfying the open set condition have large subsets close to dimension one that are coverable by Lipschitz graphs, with Lipschitz constants explicitly related to their dimensions.

Findings

Self-similar sets can have large Lipschitz graph-covered subsets near dimension one.

Lipschitz constants depend explicitly on the difference in dimensions.

Attractors of iterated function systems satisfy the open set condition and contain these subsets.

Abstract

We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph.