Continued $\mathbf{A_2}$-fractions and singular functions

Abstract

In the article we deepen the metric component of theory of infinite $A_2$-continued fractions $[0;a_1,a_2,...,a_n,...]$ with a two-element alphabet $A_2=\{\frac12,1\}$, $a_n\in A_2$ and establish the normal property of numbers of the segment $I=[\frac12;1]$ in terms of their $A_2$-representations: $x=[0;a_1,a_2,...,a_n,...]$. It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment $I$ in their $A_2$-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. This normal property of the number is effectively used to prove the singularity of the function $f(x=[0;a_1,a_2,...,a_n,...])=e^{\sum\limits_{n=1}^{\infty}(2a_n-1)v_n},$where $v_1+v_2+...+v_n+...$ is a given absolutely convergent series, when function $f$ is continuous (which is the case only if $v_n=\frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1\in R$).