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The Mysterious World of Normal Numbers

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Scalable Uncertainty Management (SUM 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9310))

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Abstract

Given an integer \(q\ge 2\), a q-normal number (or a normal number) is a real number whose q-ary expansion is such that any preassigned sequence of length \(k\ge 1\), of base q digits from this expansion, occurs at the expected frequency, namely \(1/q^k\). Even though there are no standard methods to establish if a given number is normal or not, it is known since 1909 that almost all real numbers are normal in every base q. This is one of the many reasons why the study of normal numbers has fascinated mathematicians for the past century. We present here a brief survey of some of the important results concerning normal numbers.

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References

  1. Bailey, D.H., Borwein, J.M.: Nonnormality of the Stoneham constants (2011). http://crd.lbl.gov/dhbailey/dhbpapers/nonnormality.pdf

  2. Bailey, D.H., Borwein, J.M., Crandall, R.E., Pomerance, C.: On the binary expansions of algebraic numbers. J. Number Theory Bordeaux 16, 487–518 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bailey, D.H., Crandall, R.E.: On the random character of fundamental constant expansions. Expe. Math. 10(2), 175–190 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bailey, D.H., Crandall, R.E.: Random generators and normal numbers. Exp. Math. 11(4), 527–546 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bassily, N.L., Kátai, I.: Distribution of consecutive digits in the \(q\)-ary expansions of some sequences of integers. J. Math. Sci. 78(1), 11–17 (1996)

    Article  MathSciNet  Google Scholar 

  6. Borel, E.: Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 27, 247–271 (1909)

    Article  MATH  Google Scholar 

  7. Bugeaud, Y.: Distribution Modulo One and Diophantine Approximation, Cambridge Tracts in Mathematics, vol. 193. Cambridge University Press, New York (2012)

    Book  Google Scholar 

  8. Cassels, J.W.S.: On a problem of Steinhaus about normal numbers. Colloquium Mathematicum 7, 96–101 (1959)

    MathSciNet  Google Scholar 

  9. Champernowne, D.G.: The construction of decimals normal in the scale of ten. J. London Math. Soc. 8, 254–260 (1933)

    Article  MathSciNet  Google Scholar 

  10. Copeland, A.H., Erdős, P.: Note on normal numbers. Bull. Am. Math. Soc. 52, 857–860 (1946)

    Article  MATH  Google Scholar 

  11. Davenport, H., Erdős, P.: Note on normal decimals. Can. J. Math. 4, 58–63 (1952)

    Article  MATH  Google Scholar 

  12. De Koninck, J.M., Kátai, I.: On the distribution of subsets of primes in the prime factorization of integers. Acta Arith. 72(2), 169–200 (1995)

    MATH  MathSciNet  Google Scholar 

  13. De Koninck, J.M., Kátai, I.: Construction of normal numbers by classified prime divisors of integers. Functiones et Approximatio 45(2), 231–253 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. De Koninck, J.M., Kátai, I.: On a problem on normal numbers raised by Igor Shparlinski. Bull. Aust. Math. Soc. 84, 337–349 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. De Koninck, J.M., Kátai, I.: Some new methods for constructing normal numbers. Annales des Sciences Mathématiques du Québec 36(2), 349–359 (2012)

    MATH  Google Scholar 

  16. De Koninck, J.M., Kátai, I.: Normal numbers generated using the smallest prime factor function. Annales mathématiques du Québec 38(2), 133–144 (2014)

    Article  MATH  Google Scholar 

  17. De Koninck, J.-M., Kátai, I.: The number of prime factors function on shifted primes and normal numbers. In: Rassias, T.M., Tóth, L. (eds.) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol. 94, pp. 315–326. Springer, Heidelberg (2014)

    Google Scholar 

  18. De Koninck, J.M., Luca, F.: Analytic Number Theory: Exploring the Anatomy of Integers, Graduate Studies in Mathematics, vol. 134. American Mathematical Society, Providence (2012)

    Google Scholar 

  19. Madritsch, M.G., Thuswaldner, J.M., Tichy, R.F.: Normality of numbers generated by the values of entire functions. J. Number Theory 128, 1127–1145 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Martin, G.: Absolutely abnormal numbers. Am. Math. Monthly 108(8), 746–754 (2001)

    Article  MATH  Google Scholar 

  21. Nakai, Y., Shiokawa, I.: Discrepancy estimates for a class of normal numbers. Acta Arith. 62(3), 271–284 (1992)

    MATH  MathSciNet  Google Scholar 

  22. Nakai, Y., Shiokawa, I.: Normality of numbers generated by the values of polynomials at primes. Acta Arith. 81(4), 345–356 (1997)

    MathSciNet  Google Scholar 

  23. Sierpiński, W.: Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d’un tel nombre. Bull. Soc. Math. Fr. 45, 127–132 (1917)

    Google Scholar 

  24. Stoneham, R.: On absolute \((j,\epsilon )\)-normality in the rational fractions with applications to normal numbers. Acta Arith. 22, 277–286 (1973)

    MATH  Google Scholar 

  25. Vandehey, J.: The normality of digits in almost constant additive functions. Monatsh. Math. 171(3-4), 481–497 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques et de Statistique, Université Laval, Québec, G1V 0A6, Canada

    Jean-Marie De Koninck

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  1. Jean-Marie De Koninck
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Correspondence to Jean-Marie De Koninck .

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Editors and Affiliations

  1. Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Hagen, Germany

    Christoph Beierle

  2. California Polytechnic State University, SAN LUIS OBISPO, USA

    Alex Dekhtyar

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De Koninck, JM. (2015). The Mysterious World of Normal Numbers. In: Beierle, C., Dekhtyar, A. (eds) Scalable Uncertainty Management. SUM 2015. Lecture Notes in Computer Science(), vol 9310. Springer, Cham. https://doi.org/10.1007/978-3-319-23540-0_1

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  • DOI: https://doi.org/10.1007/978-3-319-23540-0_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23539-4

  • Online ISBN: 978-3-319-23540-0

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