Fibonacci-even Numbers: Binary Additive Problem, Distribution over Progressions, and Spectrum

The representations −→ N 1 + −→ N 2 = D of a natural number D as the sum of two Fibonacci-even numbers −→ N i = F1 ◦ Ni, where ◦ 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)| ≤ 5 + ( 1 ln(1/τ) + 1 ln 2 ) lnD, where τ is the golden section. The problem concerning the distribution of Fibonacci-even numbers −→ N over arithmetic progressions −→ N ≡ r mod d is also studied. Let lF1 (d, r,X) be the number of N ’s, 0 ≤ N ≤ X, satisfying the above congruence. Then the asymptotic formula lF1 (d, r,X) = X d + c(d) lnX 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( 1/2 lnX ). The set −→ Z of Fibonacci-even numbers is an integral modification of the well-known one-dimensional Fibonacci quasilattice F . Like F , the set −→Z is a quasilattice, but it is not a model set. However, it is shown that the spectra ΛF and Λ− →Z coincide up to a scale factor ν = 1 + τ2, and an explicit formula is obtained for the structural amplitudes f− → Z (λ), where λ = a+ bτ lies in the spectrum: f− → Z (λ) = sin(πbτ) πbτ exp(−3πi bτ). Introduction 0.1. The one-dimensional Fibonacci quasilattice F. We define a map δ of the set of rational integers Z to the half-open interval J = [−1, τ ) by setting (0.1) Z δ −→ J : N → δ(N) = N − [(N + 1)τ ]τ̃ , where τ = −1+ √ 5 2 is the golden section, τ̃ = τ + 1 and [x] is the integral part of x. With the help of the map δ, we define the subset (0.2) F = δ′(Z) = {δ′(N) = N + [(N + 1)τ ]τ ; N ∈ Z} 2000 Mathematics Subject Classification. Primary 06A11.