DENSING SETS

Let H be a family of ''large'' (in various senses, e.g., of positive H ausdorff dimension or Lebesgue measure) subsets of R. We study sets D of real numbers which are H-densing, namely have the property that, gi ven any set H epsilon H and epsilon > 0, there exist an a epsilon D fo r which the set aH is epsilon-dense module 1. In the special case, whe re H consists of all subsets of R having a finite accumulations point, H-densing sets are simply Glasner sets, studied earlier. (C) 1995 Aca demic Press, Inc.