Normal Numbers and the Normality Measure
2013 ◽
Vol 22
(3)
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pp. 342-345
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Keyword(s):
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.
2014 ◽
Vol 24
(4)
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pp. 658-679
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Keyword(s):
2015 ◽
Vol 158
(3)
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pp. 419-437
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2016 ◽
Vol 164
(1)
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pp. 147-178
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Keyword(s):
2016 ◽
Vol 26
(1)
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pp. 52-67
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2017 ◽
Vol 27
(2)
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pp. 245-273
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Keyword(s):
2017 ◽
Vol 60
(3)
◽
pp. 513-525
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Keyword(s):
1953 ◽
Vol 49
(1)
◽
pp. 59-62
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2016 ◽
Vol 59
(3)
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pp. 533-547
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Keyword(s):