A Continuous digit projector from binary representations of numbers onto $A_2$-representation

Abstract

As is known, the $A_2$-continued representation of numbers is not topologically equivalent to the classical binary representation; therefore, the digit projector of such representations is a discontinuous function. In this paper, we introduce a continuous function that serves as an analogue of the digit projector of the classical binary representation of numbers into the digits of the $A_2$-continued representation with zero redundancy, namely a function of the form \[f(\Delta^2_{\alpha_1\alpha_2...\alpha_{2n-1}\alpha_{2n}...})= \Delta^{A_2}_{(\frac{1}{2})^{1-\alpha_1}(\frac{1}{2})^{\alpha_2}... (\frac{1}{2})^{1-\alpha_{2n-1}}(\frac{1}{2})^{\alpha_{2n}}...}, \alpha_n\in \{0,1\}.\] It is proved that the function $f$ is well-defined, continuous, and monotone. Using the normal properties of numbers with respect to their binary representation and Lebesgue's theorem asserting the existence of a finite derivative for a continuous monotone function almost everywhere, the singularity of the function $f$ is established. The paper also establishes a relationship between the considered function, the right-shift operator on digits, and the inversor of the continued representation of numbers. This relationship is then used to establish the singularity of the inversor.