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

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

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

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

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

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:

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