Generalised golden ratios over integer alphabets

Abstract

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