2017
Том 69
№ 6

All Issues

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

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

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Abstract

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.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 4, pp 541-555.

Citation Example: Dmytrenko S. O., Kyurchev D. V., Pratsiovytyi M. V. $A_2$-continued fraction representation of real numbers and its geometry // Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 452-463.

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