Hereditarily Non Uniformly Perfect Sets

TL;DR

This paper introduces hereditarily non uniformly perfect sets, exploring their properties and relationships with other small or porous sets, including an example with Hausdorff dimension 2 and positive capacity.

Contribution

It defines hereditarily non uniformly perfect sets and provides the first example of such a set with Hausdorff dimension 2 and positive capacity.

Findings

Hereditarily non uniformly perfect sets are distinct from porous and measure-zero sets.

An example of a hereditarily non uniformly perfect set with Hausdorff dimension 2 is constructed.

The set has positive logarithmic capacity, illustrating its size and complexity.

Abstract

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an example of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.