The Zeckendorf expansion of polynomial sequences
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 439-475.

Nous montrons que la fonction æsomme de chiffresÆ de Zeckendorf s z (n) lorsque n parcourt l’ensemble des nombres premiers ou bien une suite polynomiale d’entiers satisfait un théorème central limite. Nous obtenons aussi des résultats analogues pour d’autres fonctions du même type. Nous montrons également que le développement de Zeckendorf et le développement standard en base q des entiers sont asymptotiquement indépendants.

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.

@article{JTNB_2002__14_2_439_0,
     author = {Drmota, Michael and Steiner, Wolfgang},
     title = {The {Zeckendorf} expansion of polynomial sequences},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {439--475},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     mrnumber = {2040687},
     zbl = {1077.11005},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_2002__14_2_439_0/}
}
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
SP  - 439
EP  - 475
VL  - 14
IS  - 2
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_2002__14_2_439_0/
LA  - en
ID  - JTNB_2002__14_2_439_0
ER  - 
%0 Journal Article
%A Drmota, Michael
%A Steiner, Wolfgang
%T The Zeckendorf expansion of polynomial sequences
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 439-475
%V 14
%N 2
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_2002__14_2_439_0/
%G en
%F JTNB_2002__14_2_439_0
Drmota, Michael; Steiner, Wolfgang. The Zeckendorf expansion of polynomial sequences. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 439-475. http://archive.numdam.org/item/JTNB_2002__14_2_439_0/

[1] N.L. Bassily, I. Kátai, Distribution of the values of q-additive functions on polynomial sequences. Acta Math. Hung. 68 (1995), 353-361. | MR | Zbl

[2] J. Coquet, Corrélation de suites arithmétiques. Sémin. Delange-Pisot-Poitou, 20e Année 1978/79, Exp. 15, 12 p. (1980). | Numdam | MR | Zbl

[3] H. Delange, Sur les fonctions q-additives ou q-multiplicatives. Acta Arith. 21 (1972), 285-298. | MR | Zbl

[4] R.L. Dobrušin, Central limit theorem for nonstationary Markov chains II. Theory Prob. Applications 1 (1956), 329-383. (Translated from: Teor. Vareojatnost. i Primenen. 1 (1956), 365-425.) | MR | Zbl

[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. | MR | Zbl

[6] M. Drmota, The joint distribution of q-additive functions. Acta Arith. 100 (2001), 17-39. | MR | Zbl

[7] M. Drmota, Irregularities of Distributions with Respect to Polytopes. Mathematika, 43 (1996), 108-119. | MR | Zbl

[8] M. Drmota, M. Fuchs, E. Manstavicius, Functional Limit Theorems for Digital Expansions. Acta Math. Hung., to appear, | MR | Zbl

[9] M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer Verlag, Berlin, 1998. | MR | Zbl

[10] J.M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions. J. Theoret. Comput. Sci. 65 (1989), 153-169. | MR | Zbl

[11] J.M. Dumont, A. Thomas, Gaussian asymptotic properties of the sum-of digits functions. J. Number Th. 62 (1997), 19-38. | MR | Zbl

[12] G. Farinole, Représentation des nombres réels sur la base du nombre d'or, Application aux nombres de Fibonacci. Prix Fermat Junior 1999, Quadrature 39 (2000).

[13] P. Grabner, R.F. Tichy, α-expansions, linear recurrences and the sum-of-digits function. Manuscripta Math. 70 (1991), 311-324. | Zbl

[14] L.K. Hua, Additive Theory of Prime Numbers. Translations of Mathematical Monographs Vol. 13, Am. Math. Soc., Providence, 1965. | MR | Zbl

[15] B.A. Lifšic, On the convergence of moments in the central limit theorem for nonhomogeneous Markov chains. Theory Prob. Applications 20 (1975), 741-758. (Translated from: Teor. Vareojatnost. i Primenen. 20 (1975), 755-772.) | MR | Zbl

[16] E. Manstavicius, Probabilistic theory of additive functions related to systems of numerations. Analytic and Probabilistic Methods in Number Theory, VSP, Utrecht 1997, 413-430. | MR | Zbl

[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] M. Mendès France, Nombres normaux. Applications aux fonctions pseudo-aléatoires. J. Analyse Math. 20 (1967) 1-56. | MR | Zbl

[19] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957), 477-493. | MR | Zbl

[20] I.M. Vinogradov, The method of trigonometrical sums in the theory of numbers. Interscience Publishers, London. | Zbl

[21] M. Waldschmidt, Minorations de combinaisons tineaires de logarithmes de nombres algébriques. Can. J. Math. 45 (1993), 176-224. | MR | Zbl