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arXiv · 2102.13050

The Fractal Dimension of Product Sets

Machiel van Frankenhuijsen·Clayton Moore Williams
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Abstract

Using ultraproduct techniques we define a nonstandard Minkowski dimension which exists for all bounded sets and which has the property that $\dim(A\times B)=\dim(A)+\dim(B).$ That is, our new dimension is product-summable. To illustrate our theorem we generalize an example of Falconer's to show that the standard upper Minkowski dimension, as well as the Hausdorff dimension, are not product-summable.

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Machiel van Frankenhuijsen, Clayton Moore Williams. 2021-02-25. The Fractal Dimension of Product Sets. https://arxiv.org/abs/2102.13050

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