Functiones et Approximatio Commentarii Mathematici

Height reducing problem on algebraic integers

Shigeki Akiyama, Paulius Drungilas, and Jonas Jankauskas

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Abstract

Let $\alpha$ be an algebraic integer and assume that it is {\it expanding}, i.e., its all conjugates lie outside the unit circle. We show several results of the form $\mathbb{Z}[\alpha]=\mathcal{B}[\alpha]$ with a certain finite set $\mathcal{B}\subset\mathbb{Z}$. This property is called {\it height reducing property}, which attracted special interest in the self-affine tilings. Especially we show that if $\alpha$ is quadratic or cubic trinomial, then one can choose $\mathcal{B}= \left\{0,\,\pm 1,\,\ldots,\,\pm \left(|N(\alpha)|-1\right)\right\}$, where $N(\alpha)$ stands for the absolute norm of $\\alpha$ over $\mathbb{Q}$.

Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 105-119.

Dates
First available in Project Euclid: 25 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1348578281

Digital Object Identifier
doi:10.7169/facm/2012.47.1.9

Mathematical Reviews number (MathSciNet)
MR2987115

Zentralblatt MATH identifier
1290.11144

Subjects
Primary: 11R11: Quadratic extensions
Secondary: 11R04: Algebraic numbers; rings of algebraic integers

Keywords
expanding algebraic integer height reducing property canonical number system.

Citation

Akiyama, Shigeki; Drungilas, Paulius; Jankauskas, Jonas. Height reducing problem on algebraic integers. Funct. Approx. Comment. Math. 47 (2012), no. 1, 105--119. doi:10.7169/facm/2012.47.1.9. https://projecteuclid.org/euclid.facm/1348578281


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