Uniqueness and topological properties of number representation

• Published in: Український математичний вісник
• Date: 2004
• Main Authors: Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2004
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/124622
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Uniqueness and topological properties of number representation / O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen // Український математичний вісник. — 2004. — Т. 1, № 3. — С. 331-348. — Бібліогр.: 12 назв. — англ.

Let b be a complex number with |b| > 1 and let D be a finite subset of the complex plane C such that 0 ∊ D and card D ≥ 2. A number z is representable by the system (D, b) if z = Σajbj , where aj ∊ D. We denote by F the set of numbers which are representable by (D, b) with M = −1. The set W consists of numbers that are (D, b) representable with aj = 0 for all negative j. Let F1 be a set of numbers in F that can be uniquely represented by (D, b). It is shown that: The set of all extreme points of F is a subset of F1. If 0 ∊ F1, then W is discrete and closed. If b ∊ {z : |z| > 1}\D′, where D′ is a finite or countable set associated with D and W is discrete and closed, then 0 ∊ F1. For a real number system (D, b), F is homeomorphic to the Cantor set C iff F\F1 is nowhere dense subset of R.