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

Let pi,n, i=1,...,2n-1, be the lengths of the intervals between successive points in the Brocot sequence Fn. An asymptotic formula for the quantity σ(Fn)=Σi=1N(n)pi,nβ, which improves the previously known estimates, is obtained. Bibliography: 5 titles.