Abstract
Champernowne’s number is the best-known example of a normal number, but its digits are far from random. The sequence of nucleotides in the human X chromosome appears nonrandom in a similar way. We give a new asymptotic test of pseudorandomness, based on the law of the iterated logarithm; we call this new criterion “strong normality.” We show that almost all numbers are strongly normal and that strong normality implies normality. However, Champernowne’s number is not strongly normal. We adapt a method of Sierpiński to construct an example of a strongly normal number.
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Acknowledgements
Many thanks are due to Stephen Choi for his comments on the earlier ideas in [3]. We also give many thanks to Richard Lockhart for his help with ideas in probability. We are indebted to an anonymous referee for some extremely useful comments and criticisms.
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This paper is dedicated to Jon Borwein in celebration of his 60th birthday
Communicated By Frank Garvan.
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Belshaw, A., Borwein, P. (2013). Champernowne’s Number, Strong Normality, and the X Chromosome. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_3
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DOI: https://doi.org/10.1007/978-1-4614-7621-4_3
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