### Abstract

In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number fields F. Using earlier ideas for computing independent units by Buchmann, Pethő and later Pohst we construct sequences of suitable modules in F and special elements β contained in them. The most important ingredient in our methods is the application of the LLL-reduction procedure to the bases of those modules. For LLL-reduced bases we derive improved bounds on the sizes of the basis elements. From those bounds it is quite straightforward to show that the sequence of coefficient vectors (x_{1},...,x_{n}) of the presentation of β in the module basis becomes periodic. We can show that the approximations which we obtain are close to being optimal. Moreover, it is periodic on bases of real number fields. Thus our algorithm can be considered as a generalization, within the framework of number fields, of the continued fraction algorithm.

Original language | English |
---|---|

Pages (from-to) | 422-448 |

Number of pages | 27 |

Journal | Journal of Number Theory |

Volume | 171 |

DOIs | |

Publication status | Published - Feb 1 2017 |

### Fingerprint

### Keywords

- Continued fractions
- Diophantine approximation
- Jacobi–Perron algorithm
- LLL-algorithm

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*171*, 422-448. https://doi.org/10.1016/j.jnt.2016.07.002

**On multidimensional Diophantine approximation of algebraic numbers.** / Pethő, A.; Pohst, Michael E.; Bertók, Csanád.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 171, pp. 422-448. https://doi.org/10.1016/j.jnt.2016.07.002

}

TY - JOUR

T1 - On multidimensional Diophantine approximation of algebraic numbers

AU - Pethő, A.

AU - Pohst, Michael E.

AU - Bertók, Csanád

PY - 2017/2/1

Y1 - 2017/2/1

N2 - In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number fields F. Using earlier ideas for computing independent units by Buchmann, Pethő and later Pohst we construct sequences of suitable modules in F and special elements β contained in them. The most important ingredient in our methods is the application of the LLL-reduction procedure to the bases of those modules. For LLL-reduced bases we derive improved bounds on the sizes of the basis elements. From those bounds it is quite straightforward to show that the sequence of coefficient vectors (x1,...,xn) of the presentation of β in the module basis becomes periodic. We can show that the approximations which we obtain are close to being optimal. Moreover, it is periodic on bases of real number fields. Thus our algorithm can be considered as a generalization, within the framework of number fields, of the continued fraction algorithm.

AB - In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number fields F. Using earlier ideas for computing independent units by Buchmann, Pethő and later Pohst we construct sequences of suitable modules in F and special elements β contained in them. The most important ingredient in our methods is the application of the LLL-reduction procedure to the bases of those modules. For LLL-reduced bases we derive improved bounds on the sizes of the basis elements. From those bounds it is quite straightforward to show that the sequence of coefficient vectors (x1,...,xn) of the presentation of β in the module basis becomes periodic. We can show that the approximations which we obtain are close to being optimal. Moreover, it is periodic on bases of real number fields. Thus our algorithm can be considered as a generalization, within the framework of number fields, of the continued fraction algorithm.

KW - Continued fractions

KW - Diophantine approximation

KW - Jacobi–Perron algorithm

KW - LLL-algorithm

UR - http://www.scopus.com/inward/record.url?scp=84990866189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990866189&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2016.07.002

DO - 10.1016/j.jnt.2016.07.002

M3 - Article

VL - 171

SP - 422

EP - 448

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -