# Finite $A_2$-continued fractions in the problems of rational approximations of real numbers

• M. Pratsiovytyi Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv; Mykhailo Drahomanov Ukrainian State University, Kyiv
• Ya. Goncharenko National Pedagogical Dragomanov University
• I. Lysenko National Pedagogical Dragomanov University
• S. Ratushniak Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv; Mykhailo Drahomanov Ukrainian State University, Kyiv
Keywords: $A_2$-continued fraction, $A_2$-representation of number, $A_2$-binary numbers, cylinder, basic metric relation, convergent, normal property of number, scale-invariant set, rational approximation

### Abstract

UDC 511.7+517.5

We consider finite continued fractions whose elements are numbers  $\dfrac{1}{2}$ and $1$ (the so-called $A_2$-continued fractions): $1/a_1+1/a_2+\ldots+1/a_n=[0;a_1,a_2,\ldots,a_n],$ $a_i\in A_2=\left\{\dfrac{1}{2},1\right\}.$ We study the structure of the set $F$ of values of all these fractions and the problem of the number of representations of numbers from the segment $\left[\dfrac{1}{2};1\right]$ by fractions of this kind. It is proved that the set $F\subset\left[\dfrac{1}{3};2\right]$ has a scale-invariant structure and is dense in the segment $\left[\dfrac{1}{2};1\right]$;  the set of its elements that are greater than 1 is the set of terms of two decreasing sequences approaching 1, while the set of its elements that are smaller than $\dfrac{1}{2}$  is the set of terms of two increasing sequences approaching $\dfrac{1}{2}.$ The fundamental difference between the representations of numbers with the help of finite and infinite $A_2$-fractions is emphasized. The following hypothesis is formulated: every rational number of the segment $\left[\dfrac{1}{2};1\right]$ can be represented in the form of a finite $A_2$-continued fraction.

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Published
20.06.2023
How to Cite
Pratsiovytyi, M., Y. Goncharenko, I. Lysenko, and S. Ratushniak. “ Finite $A_2$-Continued Fractions in the Problems of Rational Approximations of Real Numbers”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 6, June 2023, pp. 849 -58, doi:10.37863/umzh.v75i6.7413.
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Research articles