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Fundam. Prikl. Mat., 2010, Volume 16, Issue 6, Pages 45–62 (Mi fpm1350)  

This article is cited in 2 scientific papers (total in 2 papers)

Hyperbolas over two-dimensional Fibonacci quasilattices

V. G. Zhuravlev

Vladimir State Pedagogical University

Abstract: 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 $m=0,1,2,…$ 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
$$ n_s(\alpha,\beta;X)\sim c_s(\alpha,\beta)\ln X\quadas\quad X\to\infty $$
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
$$ \{ \begin{aligned} &A_1^2+\overleftarrow A_1^2-2A_2\overleftarrow A_2+\overleftarrow A_2^2=F_{2s},
&\overleftarrow A_1^2-2A_1\overleftarrow A_1+A_2^2-2A_2\overleftarrow A_2+2\overleftarrow A_2^2=F_{2s-1}, \end{aligned} . $$
$0\leq A_1\leq X$, $A_2\geq0$, satisfies the asymptotic formula
$$ n_s(X)\sim\frac{c_s}{\mathrm{arcosh}(1/\tau)}\ln X\quadas\quad X\to\infty. $$
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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English version:
Journal of Mathematical Sciences (New York), 2012, 182:4, 472–483

Bibliographic databases:

UDC: 511.342

Citation: V. G. Zhuravlev, “Hyperbolas over two-dimensional Fibonacci quasilattices”, Fundam. Prikl. Mat., 16:6 (2010), 45–62; J. Math. Sci., 182:4 (2012), 472–483

Citation in format AMSBIB
\Bibitem{Zhu10}
\by V.~G.~Zhuravlev
\paper Hyperbolas over two-dimensional Fibonacci quasilattices
\jour Fundam. Prikl. Mat.
\yr 2010
\vol 16
\issue 6
\pages 45--62
\mathnet{http://mi.mathnet.ru/fpm1350}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2825516}
\elib{http://elibrary.ru/item.asp?id=20285255}
\transl
\jour J. Math. Sci.
\yr 2012
\vol 182
\issue 4
\pages 472--483
\crossref{https://doi.org/10.1007/s10958-012-0751-1}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84859496160}


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    This publication is cited in the following articles:
    1. A. A. Zhukova, A. V. Shutov, “Binarnaya additivnaya zadacha s chislami spetsialnogo vida”, Chebyshevskii sb., 16:3 (2015), 246–275  mathnet  elib
    2. V. G. Zhuravlev, “Symmetrization of bounded remainder sets”, St. Petersburg Math. J., 28:4 (2017), 491–506  mathnet  crossref  mathscinet  isi  elib
  • Фундаментальная и прикладная математика
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