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Benford Behavior of Generalized Zeckendorf Decompositions

  • Andrew Best
  • Patrick Dynes
  • Xixi Edelsbrunner
  • Brian McDonald
  • Steven J. Miller
  • Kimsy Tor
  • Caroline Turnage-Butterbaugh
  • Madeleine Weinstein
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 220)

Abstract

We prove connections between Zeckendorf decompositions and Benford’s law. Recall that if we define the Fibonacci numbers by \(F_1 = 1, F_2 = 2\), and \(F_{n+1} = F_n + F_{n-1}\), every positive integer can be written uniquely as a sum of nonadjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form \(G_{n+1}=c_1G_n+\cdots +c_LG_{n+1-L}\) with \(c_i\) positive and some other restrictions. Additionally, a set \(S \subset \mathbb {Z}\) is said to satisfy Benford’s law base 10 if the density of the elements in S with leading digit d is \(\log _{10}{(1+\frac{1}{d})}\); in other words, smaller leading digits are more likely to occur. We prove that as \(n\rightarrow \infty \) for a randomly selected integer m in \([0, G_{n+1})\) the distribution of the leading digits of the summands in its generalized Zeckendorf decomposition converges to Benford’s law almost surely. Our results hold more generally: One obtains similar theorems to those regarding the distribution of leading digits when considering how often values in sets with density are attained in the summands in the decompositions.

Keywords

Zeckendorf decompositions Fibonacci numbers Positive linear recurrence relations Benford’s law 

MSC 2010:

11B39 11B05 60F05 (primary)11K06 65Q30 62E20 (secondary) 

Notes

Acknowledgements

This research was conducted as part of the 2014 SMALL REU program at Williams College and was supported by NSF grants DMS1265673, DMS1561945, DMS1347804, Williams College, and the Clare Boothe Luce Program of the Henry Luce Foundation. It is a pleasure to thank them for their support, and the participants at SMALL and at the 16th International Conference on Fibonacci Numbers and their Applications for helpful discussions.

References

  1. 1.
    H. Alpert, Differences of multiple fibonacci numbers, Int. Electron. J. Combinat. Num. Theor. 9, 745–749 (2009)Google Scholar
  2. 2.
    O. Beckwith, A. Bower, L. Gaudet, R. Insoft, S. Li, S.J. Miller, P. Tosteson, The average gap distribution for generalized Zeckendorf decompositions. Fibonacci Quart. 51, 13–27 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    F. Benford, The law of anomalous numbers, Proc. Am. Philos. Soc. 78, 551–572 (1938), https://www.jstor.org/stable/984802
  4. 4.
    A. Berger, T. Hill, An Introduction to Benford’s Law, (Princeton University Press, 2015)Google Scholar
  5. 5.
    A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, S.J. Miller, K. Tor, C. Turnage-Butterbaugh, M. Weinstein, Benford behavior of Zeckendorf decompositions. Fibonacci Quart. 52(5), 35–46 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. Brown, R. Duncan, Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences, Fibonacci Quart. 8, 482–486 (1970), https://www.mathstat.dal.ca/FQ/Scanned/8-5/brown.pdf
  7. 7.
    M. Catral, P. Ford, P. Harris, S.J. Miller, D. Nelson, Generalizing Zeckendorf’s theorem: the Kentucky sequence. Fibonacci Quart. 52(5), 68–90 (2014)MathSciNetGoogle Scholar
  8. 8.
    M. Catral, P. Ford, P. Harris, S.J. Miller, D. Nelson, Legal decompositions arising from non-positive linear recurrences. Fibonacci Quart. 54(4), 3448–3465 (2016)MathSciNetGoogle Scholar
  9. 9.
    D.E. Daykin, Representation of natural numbers as sums of generalized fibonacci numbers. J. London Math. Soc. 35, 143–160 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Demontigny, T. Do, A. Kulkarni, S.J. Miller, D. Moon, U. Varma, Generalizing Zeckendorf’s theorem to \(f\)-decompositions. J. Num. Theor. 141, 136–158 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    P. Demontigny, T. Do, A. Kulkarni, S.J. Miller, U. Varma, A generalization of fibonacci far-difference representations and Gaussian behavior. Fibonacci Quart. 52(3), 247–273 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    M. Drmota, J. Gajdosik, The distribution of the sum-of-digits function. J. Théor. Nombrés Bordeaux 10(1), 17–32 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J.M. Dumont, A. Thomas, Gaussian asymptotic properties of the sum-of-digits function. J. Num. Theor. 62(1), 19–38 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. Filipponi, P.J. Grabner, I. Nemes, A. Pethö, R.F. Tichy, Corrigendum to: generalized Zeckendorf expansions. Appl. Math. Lett. 7(6), 25–26 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P.J. Grabner, R.F. Tichy, Contributions to digit expansions with respect to linear recurrences. J. Num. Theor. 36(2), 160–169 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    P.J. Grabner, R.F. Tichy, I. Nemes, A. Pethö, Generalized Zeckendorf expansions. Appl. Math. Lett. 7(2), 25–28 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    T. Hill, The first-digit phenomenon. Am. Scient. 86, 358–363 (1996)CrossRefGoogle Scholar
  18. 18.
    T. Hill, A statistical derivation of the significant-digit law. Statist. Sci. 10, 354–363 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. J. Keller, Generalizations of Zeckendorf’s theorem, Fibonacci Quart. 101(special issue on representations), 95–102 (1972)Google Scholar
  20. 20.
    M. Kologlu, G. Kopp, S.J. Miller, Y. Wang, On the number of summands in Zeckendorf decompositions. Fibonacci Quart. 49(2), 116–130 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    T. Lengyel, A counting based proof of the generalized Zeckendorf’s theorem. Fibonacci Quart. 44(4), 324–325 (2006)MathSciNetzbMATHGoogle Scholar
  22. 22.
    S. J. Miller (ed.), Benford’s Law: Theory and Applications, (Princeton University Press, 2015)Google Scholar
  23. 23.
    S.J. Miller, R. Takloo-Bighash, An Invitation to Modern Number Theory (Princeton University Press, Princeton, NJ, 2006)zbMATHGoogle Scholar
  24. 24.
    S.J. Miller, Y. Wang, From fibonacci numbers to central limit type theorems, J. Combinator. Theor. Series A 119 Google Scholar
  25. 25.
    S.J. Miller, Y. Wang, Gaussian Behavior in Generalized Zeckendorf Decompositions, in Combinatorial and Additive Number Theory, CANT 2011 and 2012 ed. by Melvyn B. Nathanson, Springer Proceedings in Mathematics & Statistics (2014), pp. 159–173, https://arXiv.org/pdf/1107.2718v1
  26. 26.
    S. Newcomb, Note on the frequency of use of the different digits in natural numbers. Amer. J. Math. 4, 39–40 (1881)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    R.A. Raimi, The first digit problem. Amer. Math. Monthly 83(7), 521–538 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    W. Steiner, Parry expansions of polynomial sequences, Integers 2 (2002), Paper A14Google Scholar
  29. 29.
    W. Steiner, The joint distribution of Greedy and lazy fibonacci expansions. Fibonacci Quart. 43, 60–69 (2005)MathSciNetzbMATHGoogle Scholar
  30. 30.
    L. Washington, Benford’s law for fibonacci and Lucas numbers, Fibonacci Quart. 19(2), 175–177 (1981), http://www.fq.math.ca/Scanned/19-2/washington.pdf
  31. 31.
    E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas. Bulletin de la Société Royale des Sciences de Liége 41, 179–182 (1972)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Andrew Best
    • 1
  • Patrick Dynes
    • 2
  • Xixi Edelsbrunner
    • 3
  • Brian McDonald
    • 4
  • Steven J. Miller
    • 3
  • Kimsy Tor
    • 5
  • Caroline Turnage-Butterbaugh
    • 6
  • Madeleine Weinstein
    • 7
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA
  3. 3.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  4. 4.Department of MathematicsUniversity of ChicagoChicagoUSA
  5. 5.Department of MathematicsPierre and Marie Curie University–Paris 6ParisFrance
  6. 6.Department of MathematicsDuke UniversityDurhamUSA
  7. 7.Department of MathematicsUniversity of California BerkeleyBerkeleyUSA

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