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 Algebra i Analiz, 2008, Volume 20, Issue 3, Pages 18–46 (Mi aa512)

Research Papers

Even Fibonacci numbers: the binary additive problem, the distribution over progressions, and the spectrum

V. G. Zhuravlev

Abstract: The representations $\overrightarrow{N}_1+\overrightarrow{N}_2=D$ of a natural number $D$ as the sum of two even-Fibonacci numbers $\overrightarrow{N}_i=F_1 \circ N_i$, where $\circ$ is the circular Fibonacci multiplication, are considered. For the number $s(D)$ of solutions, the asymptotic formula $s(D)=c(D)D+r(D)$ is proved; here $c(D)$ is a continuous, piecewise linear function and the remainder $r(D)$ satisfies the inequality
$$|r(D)|\leq 5+(\frac{1}{\ln 1/\tau}+\frac{1}{\ln 2})\ln D,$$
where $\tau$ is the golden section.
The problem concerning the distribution of even-Fibonacci numbers $\overrightarrow{N}$ over arithmetic progressions $\overrightarrow{N}\equiv r\;\mathrm{mod}\;d$ is also studied. Let $l_{F_1}(d,r,X)$ be the number of $N's$, $0 \leq N \leq X$, satisfying the above congruence. Then the asymptotic formula
$$l_{F_1}(d,r,X)=\frac{X}{d}+c(d)\ln X$$
is true, where $c(d)=O(d\ln d)$ and the constant in $O$ does not depend on $X$$d$$r$. In particular, this formula implies the uniformity of the distribution of the even-Fibonacci numbers over progressions for all differences $d=O(\frac{X^{1/2}}{\ln X})$.
The set $\overrightarrow{\mathbb{Z}}$ of even-Fibonacci numbers is an integral modification of the well-known one-dimensional Fibonacci quasilattice $\mathcal{F}$. Like $\mathcal{F}$, the set $\overrightarrow{\mathbb{Z}}$ is a quasilattice, but it is not a model set. However, it is shown that the spectra $\Lambda_{\mathcal{F}}$ and $\Lambda_{\overrightarrow{\mathbb{Z}}}$ coincide up to a scale factor $\nu=1+\tau^2$, and an explicit formula is obtained for the structural amplitudes $f_{\overrightarrow{\mathbb{Z}}}(\lambda)$, where $\lambda=a+b \tau$ lies in the spectrum:
$$f_{\overrightarrow{\mathbb{Z}}}(\lambda)=\frac{\sin(\pi b\tau)}{\pi b\tau}\exp(-3\pi i\;b\tau).$$

Keywords: Even-Fibonacci numbers, Fibonacci quasilattices, Fibonacci circular multiplication, Diophantine equations, spectrum

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English version:
St. Petersburg Mathematical Journal, 2009, 20:3, 339–360

Bibliographic databases:

MSC: 06A11

Citation: V. G. Zhuravlev, “Even Fibonacci numbers: the binary additive problem, the distribution over progressions, and the spectrum”, Algebra i Analiz, 20:3 (2008), 18–46; St. Petersburg Math. J., 20:3 (2009), 339–360

Citation in format AMSBIB
\Bibitem{Zhu08} \by V.~G.~Zhuravlev \paper Even Fibonacci numbers: the binary additive problem, the distribution over progressions, and the spectrum \jour Algebra i Analiz \yr 2008 \vol 20 \issue 3 \pages 18--46 \mathnet{http://mi.mathnet.ru/aa512} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2454451} \zmath{https://zbmath.org/?q=an:1206.11020} \elib{http://elibrary.ru/item.asp?id=11149937} \transl \jour St. Petersburg Math. J. \yr 2009 \vol 20 \issue 3 \pages 339--360 \crossref{https://doi.org/10.1090/S1061-0022-09-01051-6} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267497700002} 

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This publication is cited in the following articles:
1. V. V. Krasil'shchikov, “The spectrum of one-dimensional quasilattices”, Siberian Math. J., 51:1 (2010), 53–56
2. A. V. Shutov, “Trigonometricheskie summy nad odnomernymi kvazireshetkami”, Chebyshevskii sb., 13:2 (2012), 136–148
3. E. P. Davletyarova, A. A. Zhukova, A. V. Shutov, “Geometrization of Fibonacci numeration system and its applications to number theory”, St. Petersburg Math. J., 25:6 (2014), 893–907
4. Shutov A.V., “Ob odnoi additivnoi zadache s drobnymi dolyami”, Nauchnye vedomosti belgorodskogo gosudarstvennogo universiteta. seriya: matematika. fizika, 30 (2013), 111–120
5. A. V. Shutov, “Trigonometric Sums over One-Dimensional Quasilattices of Arbitrary Codimension”, Math. Notes, 97:5 (2015), 791–802
6. A. A. Zhukova, A. V. Shutov, “Binarnaya additivnaya zadacha s chislami spetsialnogo vida”, Chebyshevskii sb., 16:3 (2015), 246–275
7. E. P. Davletyarova, A. A. Zhukova, A. V. Shutov, “Geometrizatsiya obobschennykh sistem schisleniya Fibonachchi i ee prilozheniya k teorii chisel”, Chebyshevskii sb., 17:2 (2016), 88–112
8. A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245
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