Ramírez’s Problems and Fibers on Well Approximable Set of Systems of Affine Forms

We show that badly approximable matrices are exactly those that, for every inhomogeneous parameter, cannot be inhomogeneous approximated at every monotone divergent rate, which generalizes Ramírez’s result [25]. We also establish some metrical results of the fibers on well approximable set of systems of affine forms, which gives answers to three of Ramírez’s problems [25]. Furthermore, we prove that badly approximable systems are exactly those that for each monotone convergent rate ψ cannot be approximated at ψ. Moreover, we study the topological structure of the set of approximation functions.