Locally rich compact sets

TL;DR

This paper constructs compact metric spaces with highly rich tangent structures, capable of approximating any other compact space, and explores their geometric properties and dimensions.

Contribution

It introduces new examples of compact sets with universal tangent properties in Euclidean spaces, expanding understanding of local geometric complexity.

Findings

Constructed a compact space with any other as tangent at all points.

Provided examples of Euclidean compact sets with dense tangent collections.

Analyzed geometric properties and dimensions of these spaces.

Abstract

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a dense sub-set. Here the "almost all compact sets" means that the tangent collection contains a contracted image of any compact set of the cube and that the contraction ratios are uniformly bounded. In the Euclidean space, the distance of sub-sets is measured by the Hausdorff distance. Also the geometric properties and dimensions of such spaces and sets are studied.