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Transactions of the American Mathematical Society

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



Irregularities of distribution. VIII

Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 198 (1974), 1-22
MSC: Primary 10K30; Secondary 10K05
MathSciNet review: 0360504
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Abstract: If $ {x_1},{x_2} \ldots $ is a sequence in the unit interval $ 0 \leqslant x \leqslant 1$ and if $ S$ is a subinterval, write $ C(n,S)$ for the number of elements among $ {x_1}, \ldots ,{x_n}$ which lie in $ S$, minus $ n$ times the length of $ S$. For a well distributed sequence, $ C(n,S)$ as a function of $ n$ will be small. It is shown that the lengths of the intervals $ S$ for which $ C(n,S)(n = 1,2, \ldots )$ is bounded form at most a countable set.

References [Enhancements On Off] (What's this?)

  • [1] Harry Kesten, On a conjecture of Erdős and Szüsz related to uniform distribution 𝑚𝑜𝑑1, Acta Arith. 12 (1966/1967), 193–212. MR 0209253
  • [2] A. Ostrowski, Math. Miszellen. IX. Notiz sur Theorie der Diophantischen Approximationen, Jber. Deutsch. Math.-Verein. 36 (1927), 178-180.
  • [3] Wolfgang M. Schmidt, Irregularities of distribution. VI, Compositio Math. 24 (1972), 63–74. MR 0311590

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Keywords: Uniform distribution, irregularities of distribution, points in a cube
Article copyright: © Copyright 1974 American Mathematical Society