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 $\mathbb{Q}$.
Citation
Shigeki Akiyama. Paulius Drungilas. Jonas Jankauskas. "Height reducing problem on algebraic integers." Funct. Approx. Comment. Math. 47 (1) 105 - 119, September 2012. https://doi.org/10.7169/facm/2012.47.1.9
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