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

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.