Irregularities of distribution. VIII
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- by Wolfgang M. Schmidt PDF
- Trans. Amer. Math. Soc. 198 (1974), 1-22 Request permission
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
- Harry Kesten, On a conjecture of Erdős and Szüsz related to uniform distribution $\textrm {mod}\ 1$, Acta Arith. 12 (1966/67), 193–212. MR 209253, DOI 10.4064/aa-12-2-193-212 A. Ostrowski, Math. Miszellen. IX. Notiz sur Theorie der Diophantischen Approximationen, Jber. Deutsch. Math.-Verein. 36 (1927), 178-180.
- Wolfgang M. Schmidt, Irregularities of distribution. VI, Compositio Math. 24 (1972), 63–74. MR 311590
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 198 (1974), 1-22
- MSC: Primary 10K30; Secondary 10K05
- DOI: https://doi.org/10.1090/S0002-9947-1974-0360504-6
- MathSciNet review: 0360504