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The normality of digits in almost constant additive functions

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Abstract

We consider numbers formed by concatenating some of the base \(b\) digits from additive functions \(f(n)\) that closely resemble the prime divisor counting function \(\Omega (n)\). If we concatenate the last

$$\begin{aligned} \left\lceil y \frac{\log \log \log n }{\log b} \right\rceil \end{aligned}$$

digits of each \(f(n)\) in succession, then the number so created will be normal if and only if \(0 < y \le 1/2\). This provides insight into the randomness of digit patterns of additive functions after the Erdős-Kac theorem becomes ineffective.

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Notes

  1. Even though \(f_y(n)\) is already considered to be just a string of digits, we include the extra parenthesis to prevent confusion.

References

  1. Adler, R., Keane, M., Smorodinsky, M.: A construction of a normal number for the continued fraction transformation. J. Number Theory 13(1), 95–105 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Champernowne, D.G.: The construction of decimal normal in the scale of ten. J. Lond. Math. Soc. 8(3), 254–260 (1933)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Dajani, K., Kraaikamp, C.: Ergodic Theory of Numbers. The Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC (2002)

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

    Article  MATH  Google Scholar 

  7. De Konick, J.-M., Kátai, I.: Construction of normal numbers by classified prime divisors of integers. Funct. Approx. Comment. Math. 45(2), 231–253 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Delange, H., Halberstam, H.: A note on additive functions. Pac. J. Math. 7, 1551–1556 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  10. Erdős, P., Kac, M.: The Gaussian law of errors in the theory of additive number-theoretic functions. Am. J. Math. 62, 738–742 (1940)

    Article  Google Scholar 

  11. Koksma, J.F.: Diophantische Approximationen. Ergebnisse der Math., IV, 4. Berlin (1936)

  12. Levin, M., Smorodinsky, M.: Explicit construction of normal lattice configurations. Colloq. Math. 102(1), 33–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Madritsch, M.: A note on normal numbers in matrix number systems. Math. Pannon. 18(2), 219–227 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Madritsch, M.: Generating normal numbers over Gaussian integers. Acta Arith. 135(1), 63–90 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Mance, B.: Construction of normal numbers with respect to the Q-Cantor series expansion for certain Q. Acta Arith. 148(2), 135–152 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mance, B.: Normal numbers with respect to the Cantor series expansion. Thesis (Ph.D.)–The Ohio State University. pp. 290 (2010)

  18. Mance, B.: Typicality of normal numbers with respect to the Cantor series expansion. N. Y. J. Math. 17, 601–617 (2011)

    MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Shapiro, H.N.: Distribution functions of additive arithmetic functions. Proc. Natl. Acad. Sci. USA 42, 426–430 (1956)

    Article  MATH  Google Scholar 

  21. Smorodinsky, M., Weiss, B.: Normal sequences for Markov shifts and intrinsically ergodic subshifts. Israel J. Math. 59(2), 225–233 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tenenbaum, G.: Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge (1995)

  23. Tolstov, G.: Fourier Series. Second English translation. Translated from the Russian and with a preface by Richard A. Silverman. Dover Publications, Inc., New York (1976)

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Acknowledgments

The author wishes to thank Heini Halberstam for his helpful comments.

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  1. University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    J. Vandehey

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  1. J. Vandehey
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Correspondence to J. Vandehey.

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Communicated by A. Constantin.

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Vandehey, J. The normality of digits in almost constant additive functions. Monatsh Math 171, 481–497 (2013). https://doi.org/10.1007/s00605-012-0453-2

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  • DOI: https://doi.org/10.1007/s00605-012-0453-2

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