Continued \(A_2\)-fractions and singular functions (Q2691142)

This article is devoted to a new singular function defined in terms of \(A_2\)-continued fractions \( \frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}\equiv [0;a_1, a_2,\dots,a_n,\dots]\), where \(a_n\in A_2:=\{1/2,1\}\). It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment \(I=[1/2,1]\) 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. Consider quasi-exponential functions \(f\) related to the \(A_2\)-representation of numbers, defined by equality \(f(x=[0;a_1,a_2,\dots,a_n,\dots])=\exp(\sum_{n=1}^\infty(2 a_n-1)v_n)\), where \(v_1+v_2+\cdots+v_n+\cdots\) is a given absolutely convergent series. For a function \(f\), structural and functional relationships are indicated as well as necessary and sufficient conditions for continuity (which are: \(v_n=v_1(-1/2)^{n-1}\), \(v_1\in R\)) and monotonicity are found. In the case of the continuity of the function \(f\), we give the expression of its derivative and prove the singularity (the equality of the derivative to zero almost everywhere in the sense of the Lebesgue measure) using the above-mentioned normal property of numbers in terms of their \(A_2\)-representation. The relation between this new strictly monotonic singular function and the classical strictly increasing Minkowski question-mark function is indicated.