Finite A2-Continued Fractions in the Problems of Rational Approximations of Real Numbers

• Published in: Ukrainian mathematical journal 2023-11, Vol.75 (6), p.972-983
• Main Authors: Pratsiovytyi, M., Goncharenko, Ya, Lysenko, I., Ratushnyak, S.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Real numbers; Representations; Segments; Statistics
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-023-02241-3

We consider finite continued fractions whose elements are numbers 1 2 and 1 (the so-called A 2-continued fractions): 1 /a 1 +1 /a 2 + . . . +1 /a n = [0; a 1 ,a 2 , . . . ,a n ] , a i ∈ A 2 = 1 2 , 1 . 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 1 2 ; 1 by fractions of this kind. It is proved that the set F ⊏ 1 3 ; 2 has a scaleinvariant structure and is dense in the segment 1 2 ; 1 ; 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 1 2 is the set of terms of two increasing sequences approaching 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 from the segment 1 2 ; 1 can be represented in the form of a finite A 2 -continued fraction.