The best Diophantine approximations: the

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them together and give some new formulations. In contrast to our previous survey [17], this paper contains a wider number of results, especially dealing with the best Diophantine approximations. It also includes proofs or sometimes the sketches of proofs. Some applications of these results and methods to the theory of small denominators can be found in [14, 19] and [13]. §1. The best Diophantine approximations in sense of linear form. 1.1 Notation.