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 and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat. 
Tuomo Ojala, Tapio Rajala, Ville Suomala
math.CA