Given $1<\beta_1<\beta_2<2$, and $k\in\mathbb{N^+}, $ $k\geq2 $ define
$$ D_k=\{\sum_{j=0}^{k-1}A^jd_{i_j}:d_{i_j}\in\{(0,0)\,,(1,1)\}\} $$
where$ A=\left(\begin{array}{cc}\beta_1&0\\0&\beta_2\end{array}\right)$
then there exists $\delta>0$ such that for any $a\neq b\in D_k$
we have $|a-b|\geq\delta$. Here we use the Euclidean norm in $R^2$
