A combinatorial approach to binary positional number systems (Q397010)

A binary radix system for \({\mathbb R}^+\) is a pair \((B,\Gamma)\), where \(B>1\) is a real number called the base and \(\Gamma\) is a subset of all sequences with terms \(0,1\) denoted \(\{0,1\}^{\infty}\) with a ``decimal dot'' called the address space such that \(\Gamma\) is shift invariant and \(\pi:\Gamma\mapsto [0,1]\) given by \[ \pi(\omega_0\omega_1\cdots \omega_N {\bullet}\, \omega_{N+1}\cdots)=\sum_{n\geq -\infty}^N \omega_{N-n} B^n \] is bijective and strictly increasing where the order on the left is lexicographic. The above map is called the radix map. The main result of this paper is a classification of radix systems in terms of admissible pairs \((\alpha,\beta)\) which are binary strings with certain technical conditions. As a corollary, the author obtains that for all \(B\in (1,2)\) there are infinitely many binary radix systems with base \(B\). Similar results are obtained for a slightly modified notion of binary radix systems for the interval \([0,1]\).