Abstract:
For the number ns(α,β;X)ns(α,β;X) of points (x1,x2)(x1,x2) in the two-dimensional Fibonacci quasilattices F2mF2m of level m=0,1,2,…m=0,1,2,… lying on the hyperbola x21−αx22=βx21−αx22=β and such that 0≤x1≤X0≤x1≤X, x2≥0x2≥0, the asymptotic formula
ns(α,β;X)∼cs(α,β)lnXasX→∞ns(α,β;X)∼cs(α,β)lnXasX→∞
is established, the coefficient cs(α,β)cs(α,β) is calculated exactly. Using this, the following result is obtained. Let FmFm 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.
This publication is cited in the following 3 articles:
A. A. Zhukova, A. V. Shutov, “Additivnaya zadacha s k chislami spetsialnogo vida”, Materialy IV Mezhdunarodnoi nauchnoi konferentsii “Aktualnye problemy prikladnoi matematiki”. Kabardino-Balkarskaya respublika, Nalchik, Prielbruse, 22–26 maya 2018 g. Chast II, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 166, VINITI RAN, M., 2019, 10–21
V. G. Zhuravlev, “Symmetrization of bounded remainder sets”, St. Petersburg Math. J., 28:4 (2017), 491–506
A. A. Zhukova, A. V. Shutov, “Binarnaya additivnaya zadacha s chislami spetsialnogo vida”, Chebyshevskii sb., 16:3 (2015), 246–275