Abstract:
For the number ns(α,β;X) of points (x1,x2) in the two-dimensional Fibonacci quasilattices F2m of level m=0,1,2,… lying on the hyperbola x21−αx22=β and such that 0≤x1≤X, x2≥0, the asymptotic formula
ns(α,β;X)∼cs(α,β)lnXasX→∞
is established, the coefficient cs(α,β) is calculated exactly. Using this, the following result is obtained. Let Fm be the Fibonacci numbers, Ai∈N, i=1,2, and let ←Ai be the shift of Ai in the Fibonacci numeral system. Then the number ns(X) of all solutions (A1,A2) of the Diophantine system
{A21+←A21−2A2←A2+←A22=F2s,←A21−2A1←A1+A22−2A2←A2+2←A22=F2s−1, 0≤A1≤X, A2≥0, satisfies the asymptotic formula
ns(X)∼csarcosh(1/τ)lnXasX→∞.
Here τ=(−1+√5)/2 is the golden ratio, and cs=1/2 or 1 for s=0 or s≥1, respectively.