Geodesic
An analogue of Eminian's problem for the Fibonacci number system
Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 88-105.

Voir la notice de l'article provenant de la source Math-Net.Ru

Gelfond proved the uniformity of distribution of the sums of binary digits expansions of natural numbers in arithmetic progressions. Later, this result was generalized to many other numeration systems, including Fibonacci numeration system. Eminyan find an asymptotic formula for the number of natural $n$, not exceeding a given one, such that $n$ and $n+1$ have a given parity of the sum of digits of their binary expansions. Recently, this result was generalized by Shutov to the case of Fibonacci numeration system. In the paper we consider quite more general problem about the number of natural $n$, not exceeding $X$, such that $n$ and $n+l$ have a given parity of the sum of digits of their representations in Fibonacci numeration system. A method is presented that allows to obtain asymptotic formula for a given quantity for all $l$. It is based on the study of some special sums associated with the problems and recurrence relations for these sums. It is shown that for any $l$ and all variants of parity the leading term of the asymptotic is different from the expected value $\frac{X}{4}$. Als it is proved that the remainder has the order $O(\log X)$. For $l\leq 10$ constants in the leading term of asymptotic formulas are found explicitly. In the conclusion of the work, some open problems for further research are formulated.
Keywords: Fibonacci numers, Eminyan's problem, sums of digits. 
@article{CHEB_2022_23_2_a5,
     author = {A. A. Zhukova and A. V. Shutov},
     title = {An analogue of {Eminian's} problem for the {Fibonacci} number system},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {88--105},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a5/}
}
TY  - JOUR
AU  - A. A. Zhukova
AU  - A. V. Shutov
TI  - An analogue of Eminian's problem for the Fibonacci number system
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 88
EP  - 105
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a5/
LA  - ru
ID  - CHEB_2022_23_2_a5
ER  - 
%0 Journal Article
%A A. A. Zhukova
%A A. V. Shutov
%T An analogue of Eminian's problem for the Fibonacci number system
%J Čebyševskij sbornik
%D 2022
%P 88-105
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a5/
%G ru
%F CHEB_2022_23_2_a5
A. A. Zhukova; A. V. Shutov. An analogue of Eminian's problem for the Fibonacci number system. Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 88-105. http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a5/

[1] Drmota M., Gajdosik J., “The Parity of the Sum-of-Digits-Function of Generalized Zeckendorf Representations”, Fibonacci Quarterly, 36:1 (1998), 3–19 | DOI | MR | Zbl

[2] Gelfond A. O., “Sur les nombres qui ont des propriétés additives et multiplicatives données”, Acta Arithmetica, 13 (1968), 259–265 | DOI | MR | Zbl

[3] Lamberger M., Thuswaldner J. W., “Distribution properties of digital expansions arising from linear recurrences”, Mathematicf Slovaca, 53:1 (2003), 1–20 | MR | Zbl

[4] Mahler K., “The Spectrum of an Array and its Application to the Study of the Translation Properties of a Simple Class of Arithmetical Functions: Part Two On the Translation Properties of a Simple Class of Arithmetical Functions”, J. Math. and Physics, 6 (1927), 158–163 | DOI

[5] Shutov A., “On sum of digits of the Zeckendorf representations of two consecutive numbers”, Fibonacci Quarterly, 58:3 (2020), 203–207 | MR | Zbl

[6] Zeckendorf E., “Representation des nombres naturels par une soome de nombres de Fibonacci ou de nombres de Lucas”, Bull. Soc. Roy. Sci. Liege, 41 (1972), 179–182 | MR | Zbl

[7] Davletyarova E. P., Zhukova A. A., Shutov A. V., “Geometrizatsiya sistemy schisleniya Fibonachchi i ee prilozheniya k teorii chisel”, Algebra i analiz, 25:6 (2013), 1–23 | DOI | MR

[8] Karatsuba, A. A., Novak, B., “Arithmetical problems with numbers of special type”, Mathematical Notes, 66:2 (1999), 251–253 | DOI | DOI | MR | Zbl

[9] Karatsuba A. A., “Arithmetic problems in the theory of Dirichlet characters”, Russian Math. Surveys, 63:4 (2008), 641–690 | DOI | DOI | MR | Zbl

[10] Naumenko A.P., “O raspredelenii chisel s dvoichnym razlozheniem spetsialnogo vida v arifmeticheskikh progressiyakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 8:4 (2008), 34–37

[11] Naumenko A.P., “O chisle reshenii nekotorykh diofantovykh uravnenii v naturalnykh chislakh s zadannymi svoistvami dvoichnykh razlozhenii”, Chebysheskii sbornik, 12:1 (2011), 140–157 | MR | Zbl

[12] Shutov A.V., “Ob odnoi summe, svyazannoi s sistemoi schisleniya Fibonachchi”, Dalnevostochnyi matematicheskii zhurnal, 20:2 (2020), 271–275 | DOI | MR | Zbl

[13] Eminyan K.M., “Additivnye zadachi v naturalnykh chislakh s dvoichnymi razlozhenniyami spetsialnogo vida”, Chebysheskii sbornik, 12:1 (2011), 178–185 | MR | Zbl

[14] Eminyan K. M., “Asymptotic distribution law of primes of a special form”, Mathematical Notes, 100:4 (2016), 619–622 | DOI | DOI | MR | Zbl

[15] Eminyan K.M., “Ob odnoi binarnoi zadache”, Matematicheskie zametki, 60:4 (1996), 478–481 | DOI | MR | Zbl