Some remarks on the exponential separation and dimension preserving approximation for sets and measures

TL;DR

This paper extends the exponential separation condition in dimension theory of sets and measures, showing equivalence of definitions for homogeneous self-similar systems and exploring the density of certain sets and measures with specific dimensional properties.

Contribution

It weakens Hochman's exponential separation condition, introduces a modified version, and establishes the density of sets and measures with particular dimensional characteristics.

Findings

Modified ESC coincides with original for homogeneous self-similar IFS.

Certain sets with prescribed dimensions are dense in the space of compact sets.

Subsets of measures with specific dimensional properties are dense in measure spaces.

Abstract

In the dimension theory of sets and measures, a recent breakthrough happened due to Hochman, who introduced the exponential separation condition (ESC) and proved the Hausdorff dimension result for invariant sets and measures generated by similarities on the real line. Following this groundbreaking work, we make a modest contribution by weakening the condition. Further, we define the modified ESC using the convex hull of the attractor and show that for homogeneous self-similar IFS on $\mathbb{R},$ both definitions coincide. We also define some sets in the class of all nonempty compact sets using the Assouad and Hausdorff dimensions and subsets of measures in the space of Borel probability measures on $\mathbb{R}^m$ using the $L^q$ dimension and the Rajchman property, and prove their density in the respective spaces.