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Normal Numbers and the Normality Measure

Published online by Cambridge University Press:  05 April 2013

CHRISTOPH AISTLEITNER
CHRISTOPH AISTLEITNER*
Affiliation:
Department of Applied Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia (e-mail: [email protected])
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Abstract

In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira and Rödl proved that the minimal possible value of the normality measure of an N-element binary sequence satisfies

\begin{equation*} \biggl( \frac{1}{2} + o(1) \biggr) \log_2 N \leq \min_{E_N \in \{0,1\}^N} \mathcal{N}(E_N) \leq 3 N^{1/3} (\log N)^{2/3} \end{equation*}
for sufficiently large N, and conjectured that the lower bound can be improved to some power of N. In this note it is observed that a construction of Levin of a normal number having small discrepancy gives a construction of a binary sequence EN with (EN) = O((log N)2), thus disproving the conjecture above.

Keywords

Primary 68R15Secondary 11K4511K16
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 342 - 345
Copyright
Copyright © Cambridge University Press 2013

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References

[1]Aistleitner, C. On the limit distribution of the normality measure of random binary sequences. Preprint. http://arxiv.org/abs/1301.6454.Google Scholar
[2]Alon, N., Kohayakawa, Y., Mauduit, C., Moreira, C. G. and Rödl, V. (2006) Measures of pseudorandomness for finite sequences: Minimal values. Combin. Probab. Comput. 15 1–29.Google Scholar
[3]Alon, N., Kohayakawa, Y., Mauduit, C., Moreira, C. G. and Rödl, V. (2007) Measures of pseudorandomness for finite sequences: Typical values. Proc. Lond. Math. Soc. (3) 95 778–812.Google Scholar
[4]Bailey, D. H. and Crandall, R. E. (2002) Random generators and normal numbers. Experiment. Math. 11 527–546.Google Scholar
[5]Knuth, D. E. (1981) The Art of Computer Programming, Vol. 2, second edition, Addison-Wesley.Google Scholar
[6]Korobov, N. M. (1955) Numbers with bounded quotient and their applications to questions of Diophantine approximation. Izv. Akad. Nauk SSSR Ser. Mat. 19 361–380.Google Scholar
[7]Levin, M. B. (1999) On the discrepancy estimate of normal numbers. Acta Arith. 88 99–111.Google Scholar
[8]Mauduit, C. and Sárközy, A. (1997) On finite pseudorandom binary sequences I: Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82 365–377.Google Scholar
[9]Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods, Vol. 63 of CBMS–NSF Regional Conference Series in Applied Mathematics, SIAM.Google Scholar
[10]Schmidt, W. M. (1972) Irregularities of distribution VII. Acta Arith. 21 45–50.Google Scholar
[11]Wall, D. D. (1949) Normal numbers. PhD thesis, University of California, Berkeley.Google Scholar