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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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[1]Aistleitner, C. On the limit distribution of the normality measure of random binary sequences. Preprint.
[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. 15 129.
[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 778812.
[4]Bailey, D. H. and Crandall, R. E. (2002) Random generators and normal numbers. Experiment. Math. 11 527546.
[5]Knuth, D. E. (1981) The Art of Computer Programming, Vol. 2, second edition, Addison-Wesley.
[6]Korobov, N. M. (1955) Numbers with bounded quotient and their applications to questions of Diophantine approximation. Izv. Akad. Nauk SSSR Ser. Mat. 19 361380.
[7]Levin, M. B. (1999) On the discrepancy estimate of normal numbers. Acta Arith. 88 99111.
[8]Mauduit, C. and Sárközy, A. (1997) On finite pseudorandom binary sequences I: Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82 365377.
[9]Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods, Vol. 63 of CBMS–NSF Regional Conference Series in Applied Mathematics, SIAM.
[10]Schmidt, W. M. (1972) Irregularities of distribution VII. Acta Arith. 21 4550.
[11]Wall, D. D. (1949) Normal numbers. PhD thesis, University of California, Berkeley.
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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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