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- Title
Normal Numbers and the Normality Measure.
- Authors
AISTLEITNER, CHRISTOPH
- 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.
- Subjects
NORMAL numbers; GEOMETRIC series; BINARY sequences; LOGICAL prediction; MATHEMATICAL bounds; MEASUREMENT
- Publication
Combinatorics, Probability & Computing, 2013, Vol 22, Issue 3, p342
- ISSN
0963-5483
- Publication type
Article
- DOI
10.1017/S0963548313000084