  We study the so-called Gray filtration on the set of phantom maps between two
spaces. Using both its algebraic characterization and the Sullivan completion
approach to phantom maps, we generalize some of the recent results of Le,
McGibbon and Strom. We particularly emphasize on the set of phantom maps with
infinite Gray index, describing it in an original algebraic way.
  We furthermore introduce and study a natural filtration on SNT-sets (that is
sets of homotopy types of spaces having the same n-type for all n), which
appears to have the same algebraic characterization of the Gray one on phantom
maps. For spaces whose rational homotopy type is that of an H-space or a
co-H-space, we establish criteria permitting to determinate those subsets of
this filtration which are non trivial, generalizing work of McGibbon and
Moller.
  We finally describe algebraically the natural connection between phantom maps
and SNT-theory, associating to a phantom map its homotopy fiber or cofiber. We
use this description to show that this connection respect filtrations, and to
find generic examples of spaces for which the filtration on the corresponding
SNT-set consists of infinitely many strict inclusions.
