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The Fractal Dimension of Product Sets (2102.13050v2)
Published 25 Feb 2021 in math.GN
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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