Open Access
September 2012 Height reducing problem on algebraic integers
Shigeki Akiyama, Paulius Drungilas, Jonas Jankauskas
Funct. Approx. Comment. Math. 47(1): 105-119 (September 2012). DOI: 10.7169/facm/2012.47.1.9


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 $\mathbb{Q}$.


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Shigeki Akiyama. Paulius Drungilas. Jonas Jankauskas. "Height reducing problem on algebraic integers." Funct. Approx. Comment. Math. 47 (1) 105 - 119, September 2012.


Published: September 2012
First available in Project Euclid: 25 September 2012

zbMATH: 1290.11144
MathSciNet: MR2987115
Digital Object Identifier: 10.7169/facm/2012.47.1.9

Primary: 11R11
Secondary: 11R04

Keywords: canonical number system. , expanding algebraic integer , height reducing property

Rights: Copyright © 2012 Adam Mickiewicz University

Vol.47 • No. 1 • September 2012
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