On badly approximable vectors. (English) Zbl 1492.11110
The deep results of this paper were motivated by the result of 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.
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.
Reviewer: István Gaál (Debrecen)
MSC:
| 11J13 | Simultaneous homogeneous approximation, linear forms |
| 11J82 | Measures of irrationality and of transcendence |
Keywords:
Diophantine approximation; badly approximable vectors; best approximations; Diophantine exponentCitations:
Zbl 1460.11100References:
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