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

[2608.15158] A Continuous digit projector from binary representations of numbers onto $A_2$-representation Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Functional Analysis arXiv:2608.15158 (math) [Submitted on 15 Aug 2026] Title: A Continuous digit projector from binary representations of numbers onto $A_2$-representation Authors: O. O. Nikorak , S. P. Ratushniak View a PDF of the paper titled A Continuous digit projector from binary representations of numbers onto $A_2$-representation, by O. O. Nikorak and 1 other authors View PDF HTML (experimental) 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. Comments: 6 pages Subjects: Functional Analysis (math.FA) MSC classes: 26A21 (Primary) 26A30 (Secondary) Cite as: arXiv:2608.15158 [math.FA] (or arXiv:2608.15158v1 [math.FA] for this version) https://doi.org/10.48550/arXiv.2608.15158 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.31861/bmj2025.02.05 Focus to learn more DOI(s) linking to related resources Submission history From: Olena Nikorak [ view email ] [v1] Sat, 15 Aug 2026 10:22:36 UTC (8 KB) Full-text links: Access Paper: View a PDF of the paper titled A Continuous digit projector from binary representations of numbers onto $A_2$-representation, by O. O. Nikorak and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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