Ланцюгові $A_3$-дроби: основи метричної теорії

Abstract

We study a geometry of representation of numbers in terms of continued $A_3$-fractions, that is continued fractions, elements of which are acquire from a set $A_3\equiv\{s_0,s_1,s_2\}$, де $0<s_0<s_1<s_2, s_i\in\mathbb{R}$. We prove that if $s_0s_2=\frac{4}{3}$ and $s_1=(s_0+s_2)/2$ then each point of a certain interval has no more than two $A_3$-representation, and the set of points having two representations is countable, consequently, the encoding numbers system by means of a three-character alphabet, which is based on a decomposition of numbers in such continued fractions, has zero redundancy. The emphasis in the work is given to the topological-metric aspect of this representation (geometric sense of a figures, properties of the cylindrical and tail sets and so on).