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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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