For the number
n s (α
, β;
X) of points (
x 1 , x 2) in the two-dimensional Fibonacci quasilattices
\( \mathcal{F}_m^2 \) of level
m?=?0
, 1
, 2
,… lying on the hyperbola
x 1 2 ?
??αx 2 2 ?=?β and such that 0?
≤?x 1?
≤?X,
x 2?
≥?0, the asymptotic formula
$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $
is established, and the coefficient
c s (α, β) is calculated exactly. Using this, we obtain the following result. Let
F m be the Fibonacci numbers,
A i ∈
\( \mathbb{N} \),
i?=?1
, 2, and let
\( \overleftarrow {{A_i}} \) be the shift of
A i in the Fibonacci numeral system. Then the number
n s (
X) of all solutions (
A 1 , A 2) of the Diophantine system
$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $
0?
≤?A 1?
≤?X,
A 2?
≥?0, satisfies the asymptotic formula
$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $
Here
τ?=?(
?1?+?
√5)
/2 is the golden ratio, and
c s ?=?1
/2 or 1 for
s?=?0 or
s?≥?1, respectively.