Divergence points of self-similar measures and packing dimension

Abstract

Let μ be a self-similar measure in R d . A point x ∈ R d for which the limit lim r ↘ 0 log μ B ( x , r ) log r does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions.