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Mathematics of Computation

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A new multidimensional continued fraction algorithm

Authors: Jun-ichi Tamura and Shin-ichi Yasutomi
Journal: Math. Comp. 78 (2009), 2209-2222
MSC (2000): Primary 11J70; Secondary 68W25
Published electronically: January 29, 2009
MathSciNet review: 2521286
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Abstract: It has been believed that the continued fraction expansion of $ (\alpha,\beta)$ $ (1,\alpha,\beta$ is a $ {\mathbb{Q}}$-basis of a real cubic field$ )$ obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of $ (\langle\sqrt[3]{3}\rangle, \langle\sqrt[3]{9}\rangle)$ ( $ \langle x\rangle$ denoting the fractional part of $ x$). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of $ (\alpha,\beta)$ with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.

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Additional Information

Jun-ichi Tamura
Affiliation: 3-3-7-307 Azamino Aoba-ku, Yokohama, 225-0011 Japan

Shin-ichi Yasutomi
Affiliation: General Education, Suzuka National College of Technology, Shiroko Suzuka Mie 510-0294, Japan

Keywords: Diophantine approximation, multidimensional continued fraction algorithm
Received by editor(s): May 8, 2008
Received by editor(s) in revised form: August 25, 2008
Published electronically: January 29, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.