The Zeckendorf expansion of polynomial sequences

Michael Drmota; Wolfgang Steiner

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 439-475
  • ISSN: 1246-7405

Abstract

top
In the first part of the paper we prove that the Zeckendorf sum-of-digits function s z ( n ) and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a central limit theorem. We also prove that the Zeckendorf expansion and the q -ary expansions of integers are asymptotically independent.

How to cite

top

Drmota, Michael, and Steiner, Wolfgang. "The Zeckendorf expansion of polynomial sequences." Journal de théorie des nombres de Bordeaux 14.2 (2002): 439-475. <http://eudml.org/doc/248899>.

@article{Drmota2002,
abstract = {In the first part of the paper we prove that the Zeckendorf sum-of-digits function $s_z(n)$ and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a central limit theorem. We also prove that the Zeckendorf expansion and the $q$-ary expansions of integers are asymptotically independent.},
author = {Drmota, Michael, Steiner, Wolfgang},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {2},
pages = {439-475},
publisher = {Université Bordeaux I},
title = {The Zeckendorf expansion of polynomial sequences},
url = {http://eudml.org/doc/248899},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Drmota, Michael
AU - Steiner, Wolfgang
TI - The Zeckendorf expansion of polynomial sequences
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 439
EP - 475
AB - In the first part of the paper we prove that the Zeckendorf sum-of-digits function $s_z(n)$ and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a central limit theorem. We also prove that the Zeckendorf expansion and the $q$-ary expansions of integers are asymptotically independent.
LA - eng
UR - http://eudml.org/doc/248899
ER -

References

top
  1. [1] N.L. Bassily, I. Kátai, Distribution of the values of q-additive functions on polynomial sequences. Acta Math. Hung.68 (1995), 353-361. Zbl0832.11035MR1333478
  2. [2] J. Coquet, Corrélation de suites arithmétiques. Sémin. Delange-Pisot-Poitou, 20e Année 1978/79, Exp. 15, 12 p. (1980). Zbl0432.10031MR582427
  3. [3] H. Delange, Sur les fonctions q-additives ou q-multiplicatives. Acta Arith.21 (1972), 285-298. Zbl0219.10062MR309891
  4. [4] R.L. Dobrušin, Central limit theorem for nonstationary Markov chains II. Theory Prob. Applications1 (1956), 329-383. (Translated from: Teor. Vareojatnost. i Primenen. 1 (1956), 365-425.) Zbl0093.15001MR97112
  5. [5] M. Drmota, The distribution of patterns in digital expansions. In: Algebraic Number Theory and Diophantine Analysis (F. Halter-Koch and R. F. Tichy eds.), de Gruyter, Berlin, 2000, 103-121. Zbl0958.11053MR1770457
  6. [6] M. Drmota, The joint distribution of q-additive functions. Acta Arith.100 (2001), 17-39. Zbl1057.11006MR1864623
  7. [7] M. Drmota, Irregularities of Distributions with Respect to Polytopes. Mathematika, 43 (1996), 108-119. Zbl0861.11045MR1401710
  8. [8] M. Drmota, M. Fuchs, E. Manstavicius, Functional Limit Theorems for Digital Expansions. Acta Math. Hung., to appear, Zbl1026.11013MR1440487
  9. [9] M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics1651, Springer Verlag, Berlin, 1998. Zbl0877.11043MR1470456
  10. [10] J.M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions. J. Theoret. Comput. Sci.65 (1989), 153-169. Zbl0679.10010MR1020484
  11. [11] J.M. Dumont, A. Thomas, Gaussian asymptotic properties of the sum-of digits functions. J. Number Th.62 (1997), 19-38. Zbl0869.11009MR1430000
  12. [12] G. Farinole, Représentation des nombres réels sur la base du nombre d'or, Application aux nombres de Fibonacci. Prix Fermat Junior 1999, Quadrature39 (2000). 
  13. [13] P. Grabner, R.F. Tichy, α-expansions, linear recurrences and the sum-of-digits function. Manuscripta Math.70 (1991), 311-324. Zbl0725.11005
  14. [14] L.K. Hua, Additive Theory of Prime Numbers. Translations of Mathematical Monographs Vol. 13, Am. Math. Soc., Providence, 1965. Zbl0192.39304MR194404
  15. [15] B.A. Lifšic, On the convergence of moments in the central limit theorem for nonhomogeneous Markov chains. Theory Prob. Applications20 (1975), 741-758. (Translated from: Teor. Vareojatnost. i Primenen. 20 (1975), 755-772.) Zbl0426.60021MR423534
  16. [16] E. Manstavicius, Probabilistic theory of additive functions related to systems of numerations. Analytic and Probabilistic Methods in Number Theory, VSP, Utrecht1997, 413-430. Zbl0964.11031MR1653626
  17. [17] E. Manstavicius, Sums of digits obey the Strassen law. In: Proceedings of the 38-th Conference of the Lithuanian Mathematical Society, R. Ciegis et al (Eds), Technika,, Vilnius, 1997, 33-38. 
  18. [18] M. Mendès France, Nombres normaux. Applications aux fonctions pseudo-aléatoires. J. Analyse Math. 20 (1967) 1-56. Zbl0161.05002MR220683
  19. [19] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung.8 (1957), 477-493. Zbl0079.08901MR97374
  20. [20] I.M. Vinogradov, The method of trigonometrical sums in the theory of numbers. Interscience Publishers, London. Zbl1093.11001
  21. [21] M. Waldschmidt, Minorations de combinaisons tineaires de logarithmes de nombres algébriques. Can. J. Math.45 (1993), 176-224. Zbl0774.11036MR1200327

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.