Hyperbolas over two-dimensional Fibonacci quasilattices

V. G. Zhuravlev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 6, pp. 45-62.

Résumé

For the number $n_s(\alpha,\beta;X)$ of points $(x_1,x_2)$ in the two-dimensional Fibonacci quasilattices $\mathcal F^2_m$ of level $m=0,1,2,\dots$ lying on the hyperbola $x_1^2-\alpha x_2^2=\beta$ and such that $0\leq x_1\leq X$, $x_2\geq0$, the asymptotic formula $$ n_s(\alpha,\beta;X)\sim c_s(\alpha,\beta)\ln X\quad\text{as}\quad X\to\infty $$ is established, the coefficient $c_s(\alpha,\beta)$ is calculated exactly. Using this, the following result is obtained. Let $F_m$ be the Fibonacci numbers, $A_i\in\mathbb N$, $i=1,2$, 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 $$ \left\{ \begin{aligned} ^2+\overleftarrow A_1^2-2A_2\overleftarrow A_2+\overleftarrow A_2^2=F_{2s},\\ \overleftarrow A_1^2-2A_1\overleftarrow A_1+A_2^2-2A_2\overleftarrow A_2+2\overleftarrow A_2^2=F_{2s-1}, \end{aligned} \right. $$ $0\leq A_1\leq X$, $A_2\geq0$, satisfies the asymptotic formula $$ n_s(X)\sim\frac{c_s}{\mathrm{arcosh}(1/\tau)}\ln X\quad\text{as}\quad X\to\infty. $$ Here $\tau=(-1+\sqrt5)/2$ is the golden ratio, and $c_s=1/2$ or $1$ for $s=0$ or $s\geq1$, respectively.

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