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

  • S. O. Dmytrenko
  • D. V. Kyurchev
  • M. V. Pratsiovytyi


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.
How to Cite
Dmytrenko, S. O., D. V. Kyurchev, and M. V. Pratsiovytyi. “$A_2$-Continued Fraction Representation of Real Numbers and Its Geometry”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 4, Apr. 2009, pp. 452-63, https://umj.imath.kiev.ua/index.php/umj/article/view/3033.
Research articles