Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions
Abstract
In this paper, we study a numeration system with a non--integer base a>1 over the binary alphabet A={0,1}: [0;r/(a−1)]∋ x=∑n=1∞αn/an ≡ Δraα1α2…αn…, αn∈ A. We investigate the geometry of ra--representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by f(x=∑n=1∞αn/2n) =∑n=1∞αn/an, where the classical binary representation of the argument is assumed not to end with the infinite period (1). We derive a system of two functional equations satisfied by f. We prove that the function has fractal level sets and a self--affine graph. We also evaluate the integral of f over the interval [0,1].
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BibTeX
@misc{vaskevych2026some,
title = {{Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions}},
author = {Svitlana Vaskevych and Yuliia Vovk and Oleksandr Pratsiovytyi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34746},
eprint = {2609.34746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34746},
}
RIS
TY - UNPB
TI - Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions
AU - Svitlana Vaskevych
AU - Yuliia Vovk
AU - Oleksandr Pratsiovytyi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34746
UR - https://arxiv.org/abs/2609.34746
AB - In this paper, we study a numeration system with a non--integer base a>1 over the binary alphabet A={0,1}: [0;r/(a−1)]∋ x=∑_n=1^∞α_n/a^n ≡ Δ^r_a_α_1α_2…α_n…, α_n∈ A. We investigate the geometry of r_a--representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by f(x=∑_n=1^∞α_n/2^n) =∑_n=1^∞α_n/a^n, where the classical binary representation of the argument is assumed not to end with the infinite period (1). We derive a system of two functional equations satisfied by f. We prove that the function has fractal level sets and a self--affine graph. We also evaluate the integral of f over the interval [0,1].
ER -