Ukrainian Mathematical Bulletin
Volume 1, Issue 3, 2004 pp. 335-352.
Uniqueness and Topological Properties of Number RepresentationAuthors: Oleksiy Dovgoshey, Olli Martio, Vladimir Ryazanov, Matti Vuorinen
Author institution: Oleksiy Dovgoshey, Vladimir Ryazanov, Institute of Applied Mathematics and Mechanics, NAS of Ukraine, 74 Roze Luxemburg str., Donetsk, 83114, Ukraine Olli Martio, Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hallstromin katu 2b), FIN-00014 University of Helsinki, Finland Matti Vuorinen, Department of Mathematics, FIN-20014 University of Turku, Finland
Summary: Let $b$ be a complex number with $|b|>1$ and let $D$ be a finite subset of the complex plane $\mathbb C$ such that $0\in D$ and ${\rm card} \ D\> 2$. A number $z$ is representable by the system $(D, b)$ if $z=\sum\limits_{j=-\infty}^M a_j b^j$, where $a_j\in D$. 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 $a_j=0$ for all negative $j$. Let $F_1$ 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 $F_1$. If $0\in F_1$, then $W$ is discrete and closed. If $b\in \{ z : |z|>1\} \bs D'$, where $D'$ is a finite or countable set associated with $D$ and $W$ is discrete and closed, then $0\in F_1$. For a real number system $(D, b)$, $F$ is homeomorphic to the Cantor set $C$ iff $F \bs F_1$ is nowhere dense subset of $\BR$.
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