Thin And Fat Sets For Doubling Measures In Metric Spaces

We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin or fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.The paper is organized as follows. In part one,we give the setting and meaning of the problem also our conclusion. In part two, it contains some basic facts concerning doubling measures, uniform perfectness, and fat and thin sets. In part three, we give a sufficient condition for a Cantor type set being fat. Theorem1.1is proved in part four. At the end, we present a couple of examples and open questions related to our results.