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 Chebyshevskii Sb., 2017, Volume 18, Issue 4, Pages 222–245 (Mi cheb607)

Geometrization of numeration systems

A. A. Zhukova, A. V. Shutov

Abstract: We obtain geometrization theorem for numeration systems based on greedy expansions of natural numbers on denomirators of partial convergents of an arbitrary irrational $\alpha$ from the interval $(0;1)$.
More precisely, denomirators $\{ Q_i (\alpha) \}$ of partial convergents of an arbitrary irrational $\alpha \in (0; 1)$ generate Ostrowski–Zeckendorf representations of natural numbers. These representations have the form $n = \sum\limits_{i=0}^{k} z_i( \alpha, n) Q_i ( \alpha )$ with natural conditions on $z_i( \alpha, n)$ described in the terms of partial quotients $q_i(\alpha)$. In the case $\alpha =\frac{\sqrt{5}-1}{2}$ we obtain well-known Fibonacci numeration system. For $\alpha=\frac{\sqrt{g^2+4}-g}{2}$ with $g \ge 2$ corresponding expansion is called representation of natural numbers in generalized Fibonacci numeration system.
In the paper we study the sets $\mathbb{Z} ( z_0, \ldots, z_{l} )$, of natural numbers with given ending of Ostrowski–Zeckendorf representation. Our main result is the geometrization theorem, describing the sets $\mathbb{Z} ( z_0, \ldots, z_{l} )$ in the terms of fractional parts of the form $\{ n \alpha \}$. Particularly,for any admissible ending $( z_0, \ldots, z_{l} )$ there exist efffectively computable $a$, $b\in\mathbb{Z}$ such that $n \in \mathbb{Z} ( z_0, \ldots, z_{l} )$, if and only if the fractional part$\{ (n+1) i_0 (\alpha) \}$, $i_0 (\alpha) = \max \{ \alpha; 1 - \alpha \}$, lies in the segment $[ \{a \alpha \}; \{b \alpha \} ]$. This result generalizes geometrization theorems for classical and generalized Fibonacci numeration systems, proved by authors earlier.

Keywords: numeration systems, Ostrowski–Zeckendorf representation, geometrization theorem.

DOI: https://doi.org/10.22405/2226-8383-2017-18-4-221-244

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UDC: 511.43
Accepted:15.12.2017

Citation: A. A. Zhukova, A. V. Shutov, “Geometrization of numeration systems”, Chebyshevskii Sb., 18:4 (2017), 222–245

Citation in format AMSBIB
\Bibitem{ZhuShu17} \by A.~A.~Zhukova, A.~V.~Shutov \paper Geometrization of numeration systems \jour Chebyshevskii Sb. \yr 2017 \vol 18 \issue 4 \pages 222--245 \mathnet{http://mi.mathnet.ru/cheb607} \crossref{https://doi.org/10.22405/2226-8383-2017-18-4-221-244} \elib{http://elibrary.ru/item.asp?id=30042552}