  It is well known that in dimension one the set of Dirichlet improvable real
numbers consists precisely of badly approximable and singular numbers. We show
that in higher dimensions this is not the case by proving that there exist
continuum many Dirichlet improvable vectors that are neither badly approximable
nor singular. This is a consequence of a stronger statement that involves very
well approximable points. In the last section we formulate the notion of
intermediate Dirichlet improvable sets concerning approximations by rational
planes of every intermediate dimension and show that they coincide. This
naturally extends a classical theorem of Davenport and Schmidt (1969) which
states that the simultaneous form of Dirichlet's theorem is improvable if and
only if the dual form is improvable. Consequently, our main "continuum" result
is equally valid for the corresponding intermediate Diophantine sets of badly
approximable, singular and Dircihlet improvable points.
