| Suitable gauge functions are used to define three classes of fractal measures: Hausdorff measures, packing measures, and covering measures. Relationships between these fractal measures and corresponding density functions then allow us to bound these measures on fractal sets. These density functions depend not only on the fractal measures, but also on a probability measure. Given a Cantor set, the balanced probability measure is the one most naturally determined by this set, but we consider other, unbalanced probability measures as well. In Chapter 2 we study balanced density functions, first by looking at endpoints and then by studying properties of more typical points. We prove a lower density result which allows us to evaluate packing measures of Cantor sets, and we prove a weaker upper density result which allows us to evaluate covering measures of such sets subject to an additional condition. We also look at properties of intervals on which upper density is attained, and we use some of these properties to develop a algorithm for finding upper density even when this additional condition fails. We then use this algorithm to study some of the factors which tend to complicate upper density results. Finally, in Chapter 3, we take a look at unbalanced density functions and study the possibility that useful density results may be possible for some gauge function. We determine the gauge function which provides the best possible results and then show that even this gauge function proves inadequate. |
