Ram\'irez's problems and fibers on well approximable set of systems of affine forms

Abstract

We show that badly approximable matrices are exactly those that, for any inhomogeneous parameter, can not be inhomogeneous approximated at every monotone divergent rate, which generalizes Ram\'irez's result (2018). We also establish some metrical results of the fibers on well approximable set of systems of affine forms, which gives answer to two of Ram\'irez's problems (2018). Furthermore, we prove that badly approximable systems are exactly those that, can not be approximated at each monotone convergent rate {\psi}. Moreover, we study the topological structure of the set of approximation functions.