Geometrization of numeration systems

A. A. Zhukova ; A. V. Shutov

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 222-245

Résumé

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 $\left \{ Q_i (\alpha) \right \}$ 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 $\mathbb{Z} \left ( z_0, \ldots, z_{l} \right )$, of natural numbers with given ending of Ostrowski–Zeckendorf representation. Our main result is the geometrization theorem, describing the sets $\mathbb{Z} \left ( z_0, \ldots, z_{l} \right )$ in the terms of fractional parts of the form $\left \{ n \alpha \right \}$. Particularly,for any admissible ending $\left ( z_0, \ldots, z_{l} \right )$ there exist efffectively computable $a$, $b\in\mathbb{Z}$ such that $n \in \mathbb{Z} \left ( z_0, \ldots, z_{l} \right )$, if and only if the fractional part$\left \{ (n+1) i_0 (\alpha) \right \}$, $i_0 (\alpha) = \max \left \{ \alpha; 1 - \alpha \right \}$, lies in the segment $\left [ \{a \alpha \}; \{b \alpha \} \right ]$. This result generalizes geometrization theorems for classical and generalized Fibonacci numeration systems, proved by authors earlier.

Détail

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