Dirichlet is not just bad and singular

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