Partitioning of the interval $[0,1]$ induced by the Brocot sequences

Let $p_{i,n}$, $i=1,\dots,2^{n-1}$, be the lengths of the intervals between successive points in the Brocot sequence $F_n$. An asymptotic formula for the quantity $\sigma(F_n)=\sum_{i=1}^{N(n)}p_{i,n}^\beta$, which improves the previously known estimates, is obtained.
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