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

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 \[ |r(D)|\leq 5+\Bigl (\frac {1}{\ln (1/\tau )} + \frac {1}{\ln 2} \Bigr ) \ln D, \] where $\tau$ is the golden section.

The problem concerning the distribution of Fibonacci-even numbers $\overrightarrow {N}$ over arithmetic progressions $\overrightarrow {N} \equiv r \mathtt {mod} d$ 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 $c(d)=\mathit {O}(d \ln d)$ and the constant in $\mathit {O}$ 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 $d=\mathit {O}(\frac {X^{1/2}}{\ln X})$.

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