Uniform partitions of an interval

Abstract:

Let $\{ {x_n}\}$ be a sequence of numbers in $[0, 1]$; for each $n$ let ${u_0}(n), \ldots , {u_n}(n)$ be the lengths of the intervals resulting from partitioning of $[0, 1]$ by $\{ {x_1}, {x_2}, \ldots , {x_n}\}$. For $p > 1$ put ${A^{(p)}}(n) = {(n + 1)^{p - 1}}\sum \nolimits _0^n {{{[{u_j}(n)]}^p}}$; the paper investigates the behavior of ${A^{(p)}}(n)$ as $n \to \infty$ for various sequences $\{ {x_n}\}$. Theorem 1. If ${x_n} = n\theta (\bmod 1)$ for an irrational $\theta > 0$, then $\lim \inf {A^{(p)}}(n) < \infty$. However $\lim \sup {A^{(p)}} < \infty$ if and only if the partial quotients of $\theta$ are bounded (in the continued fraction expansion of $\theta$). Theorem 2 gives the exact values for $\lim \inf$ and $\lim \sup$ when $\theta = \tfrac {1} {2}(1 + \sqrt 5 )$. Theorem 3. If ${x_n}{\text {’}}s$ are random variables, uniformly distributed on $[0, 1]$, then $\lim {A^{(p)}}(n) = \Gamma (p + 1)$ almost surely.