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Normal Numbers and the Normality Measure
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- Journal:
- Combinatorics, Probability and Computing / Volume 22 / Issue 3 / May 2013
- Published online by Cambridge University Press:
- 05 April 2013, pp. 342-345
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- Article
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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
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
\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*}
