One-dimensional Fibonacci tilings

We use the $B$-operator to construct a family of Fibonacci tilings $\operatorname{Til}(\varepsilon_m)$ of the unit interval $I_0=[0,1)$ consisting of $F_{m+1}$ short and $F_{m+2}$ long elementary intervals with the ratio of the lengths equal to the golden section $\tau=\frac{1+\sqrt{5}}2$. We prove that the tilings $\operatorname{Til}(\varepsilon_m)$ satisfy a recurrence relation similar to the relation $F_{m+2}=F_{m+1}+F_m$ for the Fibonacci numbers. The ends of the elementary intervals in the tilings $\operatorname{Til}(\varepsilon_m)$ form a sequence of points $O_0$ whose derivatives $d^mO_0 = O_0 \cap [1-\tau^{-m},1)$ are sequences $O_m$ similar to the sequence $O_0$. We compute the direct $R_m(i)$ and inverse $R_{-m}(i)$ renormalizations for the sequences $O_m$. We establish a connection between our tilings and the Sturm sequence, and give some applications of the tilings $\operatorname{Til}(\varepsilon_m)$ in the theory of numbers.