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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Fibonacci-even numbers: Binary additive problem, distribution over progressions, and spectrum

Author: V. G. Zhuravlev
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 339-360
MSC (2000): Primary 06A11
Published electronically: April 6, 2009
MathSciNet review: 2454451
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Abstract: The representations $ \overrightarrow{N}_1+\overrightarrow{N}_2=D$ of a natural number $ D$ as the sum of two Fibonacci-even 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

$\displaystyle \vert r(D)\vert\leq 5+\Bigl (\frac{1}{\ln (1/\tau)} + \frac{1}{\ln 2} \Bigr ) \ln D, $

where $ \tau$ is the golden section.

The problem concerning the distribution of Fibonacci-even numbers $ \overrightarrow{N}$ over arithmetic progressions $ \overrightarrow{N} \equiv r \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

$\displaystyle l_{F_1}(d,r,X)=\frac{X}{d} + c(d) \ln X $

is true, where $ c(d)=\emph{O}(d \ln d)$ and the constant in $ \emph{O}$ does not depend on $ X$, $ d$, or $ r$. In particular, this formula implies the uniformity of the distribution of the Fibonacci-even numbers over progressions for all differences $ d=\emph{O}(\frac{X^{1/2}}{\ln X})$.

The set $ \overrightarrow{\mathbb{Z}}$ of Fibonacci-even 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:

$\displaystyle f_{\overrightarrow{\mathbb{Z}}}(\lambda)= \frac{\sin(\pi b \tau)}{\pi b \tau} \exp(-3 \pi i \; b \tau). $

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Additional Information

V. G. Zhuravlev
Affiliation: Vladimir State Pedagogical University, Av. Stroiteleǐ 11, Vladimir 600024, Russia

Keywords: Fibonacci-even numbers, Fibonacci quasilattices, Fibonacci circular multiplication, Diophantine equations, spectrum
Received by editor(s): June 5, 2007
Published electronically: April 6, 2009
Additional Notes: Supported by RFBR (grant no. 05-01-00435)
Article copyright: © Copyright 2009 American Mathematical Society

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