Some results on Lower Assouad and quantization dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914008296062976 |
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| author | Verma, Saurabh Agrawal, Ekta Dubey, Shivam |
| author_facet | Verma, Saurabh Agrawal, Ekta Dubey, Shivam |
| contents | In this paper, we first show that the collection of all subsets of \( \mathbb{R} \) having lower dimension \( γ\in [0,1] \) is dense in \( Π(\mathbb{R}) \), the space of compact subsets of \( \mathbb{R} \). Furthermore, we show that the set of Borel probability measures with lower dimension \( β\in [0, m] \) is dense in \( Ω(\mathbb{R}^m) \), 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15282 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some results on Lower Assouad and quantization dimensions Verma, Saurabh Agrawal, Ekta Dubey, Shivam Dynamical Systems 28A80, 28A78 In this paper, we first show that the collection of all subsets of \( \mathbb{R} \) having lower dimension \( γ\in [0,1] \) is dense in \( Π(\mathbb{R}) \), the space of compact subsets of \( \mathbb{R} \). Furthermore, we show that the set of Borel probability measures with lower dimension \( β\in [0, m] \) is dense in \( Ω(\mathbb{R}^m) \), 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. |
| title | Some results on Lower Assouad and quantization dimensions |
| topic | Dynamical Systems 28A80, 28A78 |
| url | https://arxiv.org/abs/2508.15282 |