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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
References
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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