The Fractal Dimension of Product Sets

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.

14 pages