On a segment partition for entropy estimation

Let $Q_n$ be a partition of the interval $[0,1]$ defines as
$$
\begin{array}{l} Q_1 =\{0,q^2,q,1\}. \\ Q_{n+1}' = qQ_n \cap q^2Q_n, \quad Q_{n+1}'' = q^2+qQ_n \cap qQ_n, \quad Q_{n+1}'''= q^2+qQ_n \cap q+q^2Q_n, \\ Q_{n+1} = Q_{n+1}'\cup Q_{n+1}'' \cup Q_{n+1}''', \end{array}
$$
where $q^2+q=1$.
The sequence $d= 1,2,1,0,1,2,1,0,1,0,1,2,1,0,1,2,1,\dots$ defines as follows.
$$
\begin{array}{l} d_1=1, \ d_2=2,\ d_4 =0; \\ d[2F_{2n}+1 : 2F_{2n+1}+1] = d[1:2F_{2n-1}+1];\\ \quad n = 0,1,2,\dots;\\ d[2F_{2n+1}+2 : 2F_{2n+1}+2F_{2n-2}] = d[2F_{2n-1}+2:2F_{2n}];\\ d[2F_{2n+1}+2F_{2n-2}+1 : 2F_{2n+1}+2F_{2n-1}+1] = d[1:2F_{2n-3}+1];\\ d[2F_{2n+1}+2F_{2n-1}+2 : 2F_{2n+2}] = d[2F_{2n-1}+2:2F_{2n}];\\ \quad n = 1,2,3,\dots;\\ \end{array}
$$
where $F_n$ are Fibonacci numbers ($F_{-1} = 0$, $F_0=F_1=1$).
The main result of this paper.
Theorem.
\begin{gather*} Q_n' = 1 - Q_n''' =\left \{ \sum_{i=1}^k q^{n+d_i}, \ k=0,1,\dots, m_n\right\}, \\ Q_n'' = 1 - Q_n'' = \left\{q^2 + \sum_{i=m_n}^k q^{n+d_i}, k=m_n-1,m_n,\dots, m_{n+1} \right\}, \end{gather*}
where $m_{2n} = 2F_{2n-2}$, $m_{2n+1} = 2F_{2n-1}+1$.