Uniformly discrete on the $\mathbb{Z}[\beta]$

Question

Is the following statement true?

Given $1<\beta<2$ define $\mathbb{Z}[\beta]$ with coefficients belonging to $0$ or $1$, then there exists $\delta$, such that for any $f\,,g\in \mathbb{Z}[\beta]$, we have $\delta\leq|f-g|$.

Answer 1

It seems that the answer is negative. Put $\beta=\frac {1+\sqrt 5}2=1.618\dots$. Then $\beta^2=\beta+1$.