A Continuous digit projector from binary representations of numbers onto A2-representation
Abstract
As is known, the A2-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 A2-continued representation with zero redundancy, namely a function of the form f(Δ2α1α2...α2n−1α2n...)= ΔA2(1/2)1−α1(1/2)α2... (1/2)1−α2n−1(1/2)α2n..., αn∈ {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.
Versions
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BibTeX
@misc{nikorak2026continuous,
title = {{A Continuous digit projector from binary representations of numbers onto $A_2$-representation}},
author = {O. O. Nikorak and S. P. Ratushniak},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.15158},
eprint = {2608.15158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15158},
}
RIS
TY - UNPB
TI - A Continuous digit projector from binary representations of numbers onto A_2-representation
AU - O. O. Nikorak
AU - S. P. Ratushniak
PY - 2026
DA - 2026/08/15
PB - arXiv
DO - 10.48550/arxiv.2608.15158
UR - https://arxiv.org/abs/2608.15158
AB - 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(Δ^2_α_1α_2...α_2n−1α_2n...)= Δ^A_2_(1/2)^1−α_1(1/2)^α_2... (1/2)^1−α_2n−1(1/2)^α_2n..., α_n∈ {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.
ER -