On thin carpets for doubling measures

TL;DR

This paper investigates the properties of thin sets for doubling and isotropic doubling measures in Euclidean spaces, establishing conditions under which certain fractal and self-affine sets are considered thin for these measures.

Contribution

It introduces new results showing that sets with Hausdorff dimension ≤ d-1 and certain self-affine sets are thin for isotropic doubling measures, and that Barański carpets are thin for doubling measures.

Findings

Sets with Hausdorff dimension ≤ d-1 are thin for isotropic doubling measures.

Self-affine sets satisfying OSCH are thin for isotropic doubling measures.

Barański carpets are thin for doubling measures.

Abstract

We study subsets of $\R^{d}$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of $\R^{d}$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also prove that a self-affine set that satisfies $OSCH$ (open set condition with holes) is thin for isotropic doubling measures. For doubling measures, we prove that Bara\'nski carpets are thin for doubling measures.