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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.11708 |
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Table of Contents:
- For a dust-like self-similar set (generated by IFSs with the strong separation condition), Elekes, Keleti and Máthé found an invariant, called `algebraic dependence number', by considering its generating IFSs and isometry invariant self-similar measures. We find an intrinsic quantitative characterisation of this number: it is the dimension over $\mathbb{Q}$ of the vector space generated by the logarithms of all the common ratios of infinite geometric sequences in the gap length set, minus 1. With this concept, we present a lower bound on the cardinality of generating IFS (with or without separation conditions) in terms of the gap lengths of a dust-like set. We also establish analogous result for dust-like graph-directed attractors on complete metric spaces. This is a new application of the ratio analysis method and the gap sequence.