This paper develops the theory of fractal dimension, introducing several definitions of and concepts related to the Minkowski and Hausdorff dimensions of a set. After reporting some results from [1], the paper explores some bounds on constants that in turn determine bounds on intersections of sets of different dimensions. Issues related to the shortcomings of this approach are discussed, in particular the fact that all the theorems hold up to a factor of \(\epsilon\) in the exponent, and how this introduces significant limitations to the scope of this paper.