On a Segment Partition for Entropy Estimation

Let \(Q_n\) be a partition of the interval \([0,1]\) defines as\(\begin{array}{l}Q_1 =\{0,q^2,q,1\}.  \\Q_{n+1}' = qQ_n \cap q^2Q_n, \ \Q_{n+1}'' = q^2+qQ_n \cap qQ_n, \ \Q_{n+1}'''= q^2+qQ_n \cap q+q^2Q_n,  \\Q_{n+1} = Q_{n+1}'\cup Q_{n+1}'' \cup Q_{n+1}''',  \end{array}\)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.\({\bf Theorem.}\\Q_n' = 1 - Q_n''' =\left \{ \sum_{i=1}^k q^{n+d_i}, \ k=0,1,\dots, m_n\right\},\\Q_n'' = 1 - Q_n'' = \left\{q^2 + \sum_{i=m_n}^k  q^{n+d_i}, \ k=m_n-1,m_n,\dots, m_{n+1} \right\},\\\)where \(m_{2n} = 2F_{2n-2}, \ m_{2n+1} = 2F_{2n-1}+1\).