标题： 有理和代数基下的数字展开 显示英文标题

标题： Digit expansions in rational and algebraic basis

摘要： 考虑满足$|\alpha| >1$的$\alpha \in \Q(i)$。 令$\D = \{0,1,\ldots,|a_0|-1\}$，其中$a_0$是$\alpha$的最小原始多项式的独立系数。 我们引入了一种在基数$\alpha$下展开复数的方法，其数字属于$\D$，我们称之为$\alpha$-展开，这推广了Akiyama、Frougny和Sakarovitch引入的有理基数数制系统，并与Steiner和Thuswaldner引入的有理自仿射瓷砖有关。 我们定义了一个算法来获得某些高斯整数的展开，并展示了关于语言的结果。 然后我们将展开扩展到所有$x \in \C$（或$x \in \R$当$\alpha = \ab \in \Q$时，有理情况将作为我们的起点），并证明它们几乎处处唯一。 我们将它们与复平面的铺砌联系起来。 我们根据高斯素数对$\Q(i)$的$p$-进完备性来描述$\alpha$-展开。 显示英文摘要

摘要： Consider $\alpha \in \Q(i)$ satisfying $|\alpha| >1$. Let $\D = \{0,1,\ldots,|a_0|-1\}$, where $a_0$ is the independent coefficient of the minimal primitive polynomial of $\alpha$. We introduce a way of expanding complex numbers in base $\alpha$ with digits in $\D$ that we call $\alpha$-expansions, which generalize rational base number systems introduced by Akiyama, Frougny and Sakarovitch, and are related to rational self-affine tiles introduced by Steiner and Thuswaldner. We define an algorithm to obtain the expansions for certain Gaussian integers and show results on the language. We then extend the expansions to all $x \in \C$ (or $x \in \R$ when $\alpha = \ab \in \Q$, the rational case will be our starting point) and show that they are unique almost everywhere. We relate them to tilings of the complex plane. We characterize $\alpha$-expansions in terms of $p$-adic completions of $\Q(i)$ with respect to Gaussian primes.