Finite beta-expansions of natural numbers

Let \(\beta>1\). For \(x \in [0,\infty)\), we have so-called a beta-expansion of \(x\) in base \(\beta\) as follows:

$$x= \sum_{j \leq k} x_{j}\beta^{j} = x_{k}\beta^{k}+ \cdots + x_{1}\beta+x_{0}+x_{-1}\beta^{-1} + x_{-2}\beta^{-2} + \cdots$$

where \(k \in \mathbb{Z}\), \(\beta^{k} \leq x < \beta^{k+1}\), \(x_{j} \in \mathbb{Z} \cap [0,\beta)\) for all \(j \leq k\) and \(\sum_{j \leq n}x_{j}\beta^{j}<\beta^{n+1}\) for all \(n \leq k\). In this paper, we give a sufficient condition (for \(\beta\)) such that each element of \(\mathbb{N}\) has a finite beta-expansion in base \(\beta\). Moreover we also find a \(\beta\) with this finiteness property which does not have positive finiteness property.