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.
