The classical problem of metric number theory is the search for properties that hold almost everywhere in the sense of the Lebesgue measure for the corresponding representation. An important result in the corresponding context is the well-known result of E. Borel for the classical $s$-adic representation of real numbers, which was subsequently significantly developed and deepened, in particular, by means of the theory of probability and dynamical systems. The rapid development of the theory of Diophantine approximations is associated, in particular with the classical continued fraction representation as a means of finding the best, in some sense, approximations of irrational numbers to rational ones. The latter aspect undoubtedly created the need to find metric properties for numbers given by continued fractions with natural elements. The first metric results for the classical continued fraction representation were obtained by O. Khinchyn, by rather complex methods, on the basis of a thorough analysis of the topological-metric properties of the corresponding
representation. Other metric results were subsequently obtained by P.Levi. An important step in the development of methods of metric number theory was played by the theory of dynamical systems, in particular, the Birkhoff-Khinchin theorem. A relatively new concept is  the continued fraction $A_2$-representation of real numbers. This representation involves the use of only two positive digits of the alphabet, the product of which is equal to 0.5. Thus, the continued fraction $A_2$-representation of real numbers occupies an intermediate place between the classical binary representation and the classical chain one, which in turn is characterized by an infinite alphabet.
The corresponding similarity, on the other hand, does not allow using the means of the laws of large numbers  to find metric properties of numbers that have a continued fraction $A_2$-representation. It should also be noted that the direct methods used by O. Khinchin and P. Levi are also difficult to obtain the corresponding results for the continued fraction $A_2$-representation.
The paper considers some metric results given in terms of the continued fraction $A_2$-representation of real numbers with the alphabet $\{\alpha_{1};\alpha_{2}\}$. The ergodic properties of the dynamical system are studied of the system corresponding to the Bernoulli shift of the symbols of the corresponding $A_{2}$-representation of real numbers of the segment $[\alpha_{1};\alpha_{2}]$.

[1] Pratsiovytyi M.V. Two-character encoding systems for real numbers and their applications. Kyiv: Scientific Opinion, 316 (2022).
[2] Dajani K., Kraaikamp C. Ergodic theory of numbers. Mathematical Association of America,Washington, DC, 190 (2002).
[3] Dinaburg E, Sinai Ya. Statistics of solutions of the integral equation $ax-by = ±1$. Funct. Anal., Appl 1990. 24,№3, 1–13.
[4] Dmytrenko S., Kyurchev D., Pratsiovytyi M. Continued fraction $A_2$ representation of real numbers and its geometry. Ukr. Math. J 2009,61(4), 452–463. (in Ukrainian) DOI:10.1007/s11253-009-0236-7
[5] Knopp K. Mengentheoretische Behandlung einiger Probleme der diophantischen-Approximationen und der transfiniten Wahrscheinlichkeiten. Math. Annalen 1926, 85, 409–426.
[6] Khinchin A. Ya. Continued fractions. Chicago: The University of Chicago Press, 112 (1964).
[7] Khintchine A. Metrische Kettenbruchprobleme. Compositio Math. France 1935, 1, 361–382.
[8] Levy P. Sur les lois de probabilite dont dependent les quotients complets et incomplets dune fraction continue. Bull. Soc. Math. France 1929, 57, 178–194. DOI: 10.24033/bsmf.1150
[9] Pratsiovytyi M., Kyurchev D. Singularity of the distribution of a random variable represented by a continued $A_2$-fraction with independent elements. Teor. Imovir. ta Matem. Statyst 2009, 81, 139–154. (in Ukrainian)
[10] Pratsiovytyi M., Kyurchev D. Properties of the distribution of the random variable defined by $A_2$- continued fraction with independent elements. Random Oper. Stochastic Equations 2009, 17(1), 91–101. DOI:10.1515/ROSE.2009.006
[11] Ryll-Nardzewski C. On the ergodic theorems II (Ergodic theory of continued fractions). Studia Mathematica 1951, 12, 74–79.
[12] Seidel L. Untersuchungen $\ddot{u}$ber die Konvergenz und Divergenz der Kettenbr$\ddot{u}$che. Habilschrift, M$\ddot{u}$nchen (1846).
[13] Stern M. $\ddot{U}$ber die Kennzeichen der Konvergenz eines Kettenbruchs. Journal f$\ddot{u}$r die Reine und Angewandte Mathematik 1848, 37, 255–272.