A_2-continued fraction representation of real numbers and its geometry - Dmytrenko, S. O.; Kyurchev, D. V.; Pratsiovytyi, M. V. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
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.