Some results on Lower Assouad and quantization dimensions

Saurabh Verma; Ekta Agrawal; Shivam Dubey

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arXiv · 2508.15282

Some results on Lower Assouad and quantization dimensions

Saurabh Verma·Ekta Agrawal·Shivam Dubey

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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.

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Saurabh Verma, Ekta Agrawal, Shivam Dubey. 2025-08-21. Some results on Lower Assouad and quantization dimensions. https://arxiv.org/abs/2508.15282

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