## Uniform partitions of an interval

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- by Vladimir Drobot PDF
- Trans. Amer. Math. Soc.
**268**(1981), 151-160 Request permission

## 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*.

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## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**268**(1981), 151-160 - MSC: Primary 10K30; Secondary 10K05
- DOI: https://doi.org/10.1090/S0002-9947-1981-0628451-3
- MathSciNet review: 628451