# $A_2$-continued fraction representation of real numbers and its geometry

### Abstract

We study the geometry of representations of numbers by continued fractions whose elements belong to the set $A_2 = {α_1, α_2}$ ($A_2$-continued fraction representation). It is shown that, for $α_1 α_2 ≤ 1/2$, every point of a certain segment admits an $A_2$-continued fraction representation. Moreover, for $α_1 α_2 = 1/2$, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose $A_2$-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its $A_2$-continued fraction representation form a homogeneous Markov chain are also investigated.
Published

25.04.2009

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 61, no. 4, Apr. 2009, pp. 452-63, https://umj.imath.kiev.ua/index.php/umj/article/view/3033.

Issue

Section

Research articles