On badly approximable vectors (Q2137862)

The deep results of this paper were motivated by the result of \textit{N. A. Van Nguyen} et al. [J. Théor. Nombres Bordx. 32, No. 2, 387--402 (2020; Zbl 1460.11100)] where a powerful method was introduced. The authors describe badly approximable numbers, their relation with the continued fraction algorithm and also intruduce the concept in higher dimensions. A criterion is given for a vector \(\alpha\in\mathbb R^d\) to be a badly approximable vector. The authors show that a more general version of their criterion is not valid. A corresponding new result is also announced on best approximations in the sense of a linear form. In Section 4, propositions on the ordinary Diophantine exponent and the uniform Diophantine exponent are formulated.