Diophantine approximations of linear forms (Q2122467)

The present paper deals with Diophantine approximations of linear combinations of real algebraic numbers of an arbitrary degree. The author gives the following brief description of the presented results: ``An infinite sequence of integral approximations of the linear forms is generated using recurrence relation. It is proved that the resulting Diophantine approximations are the best with respect to some polyhedral norms that are ray functions or the Minkowski functionals.'' The special attention is given to explanations of the main results of this paper, to the consideration of the main notions, and to certain statements related to these notions. One can note that the following notions are considered: -- Units of an algebraic field; it includes basic units, Pisot units, and localized Pisot units. -- Pisot modular matrices; the consideration includes such notions as a complete module over the ring, the representation matrix, and the transition matrix, as well as modular and unimodular matrices, the localized Pisot matrix, etc. -- The linear form estimation contains descriptions of the decomposition of a modular Pisot matrix, of iterations of certain integer vectors, and of a certain linear form, etc. -- Linear unimodular transformations, an unimodular basic simplex, and a supersimplex, etc. -- Polyhedral norms and the best Diophantine approximations include the notions of the basic parallelepiped, of a certain real linear form, and of Minkowski functionals, etc.; the geometry of the best Diophantine approximations is also considered.