On partitions of the unit interval generated by Brocot sequences

Let $p_{i, n}, i=1,..., N(n)=2^{n-1}$ be the lengths of intervals between the neighboring fractions of Brocot sequence $F_n$. We obtain an asymptotic formula for $\sigma_{\beta}(F_n)=\sum_{i=1}^{N(n)} p_{i, n}^{\beta}$,which improves known estimates.