On partitions of the unit interval generated by Brocot sequences

Abstract

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.

36 pages