Generalised golden ratios over integer alphabets

It is a well known result that for $β\in(1,\frac{1+\sqrt{5}}{2})$ and $x\in(0,\frac{1}{β-1})$ there exists uncountably many $(ε_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ such that $x=\sum_{i=1}^{\infty}ε_{i}β^{-i}.$ When $β\in(\frac{1+\sqrt{5}}{2},2]$ there exists $x\in (0,\frac{1}{β-1})$ for which there exists a unique $(ε_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ 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.

No reviews yet.

No comments yet.