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arXiv · 2006.14366

On the intermediate dimensions of Bedford-McMullen carpets

István Kolossváry
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Abstract

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta}\Lambda$, interpolates between its Hausdorff and box dimensions using the parameter $\theta\in[0,1]$. Determining a precise formula for $\dim_{\theta}\Lambda$ is particularly challenging when $\Lambda$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this direction, answering a question of Fraser, we show that $\dim_{\theta}\Lambda$ is strictly less than the box dimension of $\Lambda$ for every $\theta<1$, moreover, the derivative of the upper bound is strictly positive at $\theta=1$. We also improve on the lower bound obtained by Falconer, Fraser and Kempton.

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István Kolossváry. 2020-06-25. On the intermediate dimensions of Bedford-McMullen carpets. https://arxiv.org/abs/2006.14366

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