The computation of the Hausdorff measure is extremely difficult due to the infinum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult or impossible. However, there are some good bounds, given by Baoguo Jia, for some exactly self-similar fractals such as the Siernpinski Triangle and Koch Curve.
One exception to this trend are the cantor sets. Their Hausdorff measure has been explicitly computed.
My question is this. Are there any other non-trivial fractals that have known exact Hausdorff measures? Could any of these other fractals be used to help aid in the calculation of other Hausdorff measures?
Edit: I'm still looking for answers, but now I know that techniques that can link the Hausdorff measure of specific fractals to more general ones will have very sophisticated arguments.