Some results on Lower Assouad and quantization dimensions

TL;DR

This paper investigates the density of sets with specific lower dimensions in the space of compact subsets and measures, explores the relationship between quantization and lower dimensions, and computes the lower dimension of an invariant measure for a product IFS.

Contribution

It establishes the density of subsets and measures with given lower dimensions, links quantization and lower dimensions under convolution, and calculates the lower dimension of a particular invariant measure.

Findings

Density of sets with fixed lower dimension in compact subsets of R

Density of measures with fixed lower dimension in probability measures on R^m

Equality of quantization and lower dimension for convoluted measures

Abstract

In this paper, we first show that the collection of all subsets of \( \mathbb{R} \) having lower dimension \( \gamma \in [0,1] \) is dense in \( \Pi(\mathbb{R}) \), the space of compact subsets of \( \mathbb{R} \). Furthermore, we show that the set of Borel probability measures with lower dimension \( \beta \in [0, m] \) is dense in \( \Omega(\mathbb{R}^m) \), the space of Borel probability measures on \( \mathbb{R}^m \). We also prove that the quantization and the lower dimension of a measure \( \vartheta \) coincide with those of the convolution of \( \vartheta \) with a finite combination of Dirac measures. In the end, we compute the lower dimension of the invariant measure associated with the product IFS.