Minkowski and packing Dimension comparisons for sets with Reifenberg properties

In Koeller \cite{koerprops} the twelve variants of the Reifenberg properties known to be instrumental in the theory of minimal surfaces were classified with respect to various Hausdorff measure based measure theoretic properties. The classification lead to the consideration of fine geometric properties and a connection to fractal geometry. The current work develops this connection and extends the classification to consider Minkowski-dimension, packing dimension, measure, and rectifiability, and the equality of packing and Hausdorff measures with interesting results.