Finite beta-expansions of natural numbers

Let \(\beta>1\) $\beta >1$ . For \(x \in [0,\infty)\) $x\in [0,\infty )$ , we have so-called a beta-expansion of \(x\) $x$ in base \(\beta\) $\beta$ as follows: \(x= \sum_{j \leq k} x_{j}\beta^{j} = x_{k}\beta^{k}+ \cdots + x_{1}\beta+x_{0}+x_{-1}\beta^{-1} + x_{-2}\beta^{-2} + \cdots\) $$x=\sum _{j\le k}{x}_j{\beta }^j=x_k{\beta }^k+\cdots +x_1\beta +x_0+x_{-1}{\beta }^{-1}+x_{-2}{\beta }^{-2}+\cdots$$ where \(k \in \mathbb{Z}\) $k\in \mathbb{Z}$ , \(\beta^{k} \leq x < \beta^{k+1}\) ${\beta }^k\le x<{\beta }^{k+1}$ , \(x_{j} \in \mathbb{Z} \cap [0,\beta)\) $x_j\in \mathbb{Z}\cap [0,\beta )$ for all \(j \leq k\) $j\le k$ and \(\sum_{j \leq n}x_{j}\beta^{j}<\beta^{n+1}\) ${\sum }_{j\le n}{x}_j{\beta }^j<{\beta }^{n+1}$ for all \(n \leq k\) $n\le k$ . In this paper, we give a sufficient condition (for \(\beta\) $\beta$ ) such that each element of \(\mathbb{N}\) $\mathbb{N}$ has a finite beta-expansion in base \(\beta\) $\beta$ . Moreover we also find a \(\beta\) $\beta$ with this finiteness property which does not have positive finiteness property.