Finite beta-expansions of natural numbers

Abstract: Let $\beta>1$. For $x \in [0,\infty)$, we have so-called the $\beta$-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 the finite beta-expansion in base $\beta$. Moreover we also find a $\beta$ with this finiteness property which does not have positive finiteness property.