Densing sets

Let H be a family of "large" (in various senses, e.g., of positive Hausdorff dimension or Lebesgue measure) subsets of R. We study sets D of real numbers which are H-densing, namely have the property that, given any set H ∈ H and ε(lunate) > 0, there exist an a ∈ D for which the set aH is ε(lunate)-dense modulo 1. In the special case, where H consists of all subsets of R having a finite accumulations point, H-densing sets are simply Glasner sets, studied earlier.