Generalised golden ratios over integer alphabets

It is a well known result that for $\beta \in (1,\frac{1+\sqrt{5}}{2})$ and $x\in (0,\frac{1}{\beta -1})$ there exists uncountably many $({ϵ}_i{)}_{i=1}^{\mathrm{\infty }}\in {0,1}^{\mathbb{N}}$ such that $x={\sum }_{i=1}^{\mathrm{\infty }}{ϵ}_i{\beta }^{-i}\mathrm{.}$ When $\beta \in (\frac{1+\sqrt{5}}{2},2]$ there exists $x\in (0,\frac{1}{\beta -1})$ for which there exists a unique $(\epsilon_{i}){i=1}^{\infty}\in {0,1}^{$\mathbb{N}$}$suchthat$x=\sum{i=1}^{\infty}\epsilon_{i}\beta^{-i}.$ In this paper we consider the more general case when our sequences are elements of ${0,\mathrm{.}\mathrm{.}\mathrm{.},m}^{\mathbb{N}}\mathrm{.}$ We show that an analogue of the golden ratio exists and give an explicit formula for it.