2019
Том 71
№ 5

All Issues

Kyurchev D. V.

Articles: 1
Article (Ukrainian)

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

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

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 452-463

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.