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

V. G. Zhuravlev

Algebra i analiz, Tome 20 (2008) no. 3, pp. 18-46.

Résumé

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 $$ |r(D)|\leq 5+\Bigl(\frac{1}{\ln 1/\tau}+\frac{1}{\ln 2}\Bigr)\ln D, $$ where $\tau$ is the golden section. The problem concerning the distribution of 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: $$ f_{\overrightarrow{\mathbb{Z}}}(\lambda)=\frac{\sin(\pi b\tau)}{\pi b\tau}\exp(-3\pi i\;b\tau). $$

Détail

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

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