##plugins.themes.bootstrap3.article.main##
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.
##plugins.themes.bootstrap3.article.details##
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.
Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.