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Fibonacci-even numbers: binary additive problem, distribution over progressions, and spectrum. (English. Russian original) Zbl 1206.11020
St. Petersbg. Math. J. 20, No. 3, 339-360 (2009); translation from Algebra Anal. 20, No. 3, 18-46 (2008).
Summary: 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 $$\displaystyle | r(D)|\leq 5+\Bigl (\frac{1}{\ln (1/\tau)} + \frac{1}{\ln 2} \Bigr ) \ln D,$$ where $$\tau$$ is the golden section. The problem concerning the distribution of Fibonacci-even numbers $$\overrightarrow{N}$$ over arithmetic progressions $$\overrightarrow{N} \equiv r (d)$$ 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 $$X$$, $$d$$, or $$r$$. In particular, this formula implies the uniformity of the distribution of the Fibonacci-even numbers over progressions for all differences $$d=O(\frac{X^{1/2}}{\ln X})$$. 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: $f_{\overrightarrow{\mathbb{Z}}}(\lambda)= \frac{\sin(\pi b \tau)}{\pi b \tau} \exp(-3 \pi i \; b \tau).$

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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