On multidimensional Diophantine approximation of algebraic numbers

A. Pethő, Michael E. Pohst, Csanád Bertók

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 (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.

Original language English 422-448 27 Journal of Number Theory 171 https://doi.org/10.1016/j.jnt.2016.07.002 Published - Feb 1 2017

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Diophantine Approximation
Algebraic number
Number field
Module
Simultaneous Approximation
Linear Forms
Dual Problem
Continued fraction
Unit
Computing
Coefficient
Approximation

Keywords

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

ASJC Scopus subject areas

• Algebra and Number Theory

Cite this

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

In: Journal of Number Theory, Vol. 171, 01.02.2017, p. 422-448.

Research output: Contribution to journalArticle

Pethő, A. ; Pohst, Michael E. ; Bertók, Csanád. / On multidimensional Diophantine approximation of algebraic numbers. In: Journal of Number Theory. 2017 ; Vol. 171. pp. 422-448.
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