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Normal Numbers and the Normality Measure


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.

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[3] N. Alon , Y. Kohayakawa , C. Mauduit , C. G. Moreira and V. Rödl (2007) Measures of pseudorandomness for finite sequences: Typical values. Proc. Lond. Math. Soc. (3) 95 778812.

[4] D. H. Bailey and R. E. Crandall (2002) Random generators and normal numbers. Experiment. Math. 11 527546.

[9] H. Niederreiter (1992) Random Number Generation and Quasi-Monte Carlo Methods, Vol. 63 of CBMS–NSF Regional Conference Series in Applied Mathematics, SIAM.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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