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arXiv · 2608.22000

Sub-self-similar sets with distinct Hausdorff and Box dimensions

Junda Zhang
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Abstract

We construct an IFS consisting of three contracting similarities on $\mathbb{R}^{9}$ with a countable compact sub-self-similar set $E$ with distinct Hausdorff and lower box dimensions. This provides a concrete counterexample to the question raised by Falconer concerning the equality of Hausdorff and box dimensions for sub-self-similar sets without the open set condition.

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Junda Zhang. 2026-08-22. Sub-self-similar sets with distinct Hausdorff and Box dimensions. https://arxiv.org/abs/2608.22000

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