Let $j\in \{1,2\}$. For each $j$, assume that $(a^j_n)_{n\ge0}\subset\{0,1\}^\infty$ is a sequence with $a^j_n\in\{1,0\}$ for each $n$. Choose $\xi_1,\xi_2\in (0,\frac{1}{2})$. Let $ d^i_n$ be the numbers:
$$ d^{i}_n = (1-\xi_i)\sum_{k=0}^n a_k\, \xi^k_i$$
Assume $\xi_1^n \neq \xi_2^m$ for any $n,m\in\mathbb N$, and $d^i_n>0$. Is it true then that $ d^1_n \neq d^2_m $ for any $n,m$? Does this relationship hold in the limits of $n$ or $m$?
Intuition: I have a strong feeling this is true: it’s a bit like saying $\sum^n_{k\ge1} 3^{-k} \neq \sum^m_{k\ge1} 4^{-k}$ for any $n,m\ge1$.
Motivation: in an answer to another question, it was pointed out to me how to characterise the elements of $\xi$-Cantor-like sets when $\xi \neq \tfrac{1}{3}$. The $\xi$-Cantor-like set for $\xi \in (0,\frac{1}{2})$ is built by iterating the transformation $[0,1] \to [0,\xi]\cup [1-\xi, 1]$. Proving the above statement will enable me to prove these Cantor-like sets are pairwise disjoint when $\xi^n \neq \eta^m$ for any $n,m\in\mathbb N$.