The Fractal Dimension of Product Sets

TL;DR

This paper introduces a new nonstandard Minkowski dimension that is defined for all bounded sets and satisfies the property that the dimension of a product set equals the sum of the dimensions, unlike traditional dimensions.

Contribution

The paper develops a nonstandard Minkowski dimension using ultraproduct techniques that is universally defined and product-summable, contrasting with standard dimensions.

Findings

The new dimension exists for all bounded sets.

It satisfies the property that im(A imes B) = im(A) + im(B).

Standard Minkowski and Hausdorff dimensions are not product-summable.

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.