Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet
Abstract
In this work, we study a numeral system with a natural base s ≥ 2 and a redundant alphabet Ar = {0, 1, ..., r}, where s ≤ r ≤ 2s - 2. We investigate the topological, metric, and fractal properties of the set of numbers in the interval [0, r/(s-1)] that admit a unique representation x = ∑n=1∞ (αn / sn) ≡ Δrsα1α2...αn..., αn ∈ Ar. A criterion for the uniqueness of a number's representation is established. We prove that the Hausdorff-Besicovitch dimension of the set of numbers with a unique representation is equal to ln(2s - r - 1) / ln s. An analysis of the quantity of representations for numbers having purely periodic representations with a simple (single-digit) period is carried out. It is proved that the set of numbers that admit a continuum of distinct representations has full Lebesgue measure. Conditions for a number to belong to this set are given in terms of one of its representations.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
semigroups and automata theory
Advanced Mathematical Theories and Applications
Publication details
- Journal / source
- Proceedings of the International Geometry Center
- Publisher
- Odesa National University of Technology
- Volume
- 19
- Issue
- 1
- Pages
- 6–6
- Publication date
- 17 May 2026
- Language
- UK