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

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.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Theorem 2.5
• Definition 2.6
• Example 2.7
• Lemma 3.1
• proof
• Theorem 3.2