We study the geometry of representations of numbers by continued fractions whose elements belong to the set A 2. It is shown that, for α α 1 2 1 2 ≤ / , every point of a certain segment admits an A 2 -continued fraction representation.Moreover, for α α 1 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.

Abstract. We study the properties of the distribution of the random variablewhere η k are independent random variables such thatIt is proved that the distribution of ξ cannot be absolutely continuous. We find the criteria for the distribution of ξ to belong to one of the two types of singular distributions, Cantor and Salem types, depending on topological and metric properties of the topological support of the distribution.

We study the distribution of a random variable ξ = 1where η k are independent random variables having the distributions:We prove that the r.v. ξ has either pure discrete (atomic) or pure continuous distribution. In the case of discreteness of the r.v. ξ, we describe the set of all its atoms. For the continuously distributed r.v. ξ, we give the formula for the distribution function and prove the criterion for singularity of Cantor type.Key words. Random variable; random continued fraction with independent elements; A 2 -continued fraction; purity of the probability distribution; singularly continuous probability distribution of Cantor type.