Three-dimensional Symmetric Shift Radix Systems

Abstract

Shift radix systems have been introduced by Akiyama et al. as a common generalization of β-expansions and canonical number systems. In the present paper we study a variant of them, so-called symmetric shift radix systems which were introduced recently by Akiyama and Scheicher. In particular, for d ∈ N and r ∈ R let τr : Z d → Z , a = (a1, . . . , ad) 7→ a2, . . . , ad,− r1a1 + r2a2 + · · · + rdad + 1 2 . The mapping τr is called a symmetric shift radix system, if ∀a ∈ Z d ∃n ∈ N : τ r (a) = 0. Akiyama and Scheicher found out that the parameters r giving rise to a symmetric shift radix system in R form an isosceles triangle together with parts of its boundary. In the present paper we completely characterize all symmetric shift radix systems in the three dimensional space. The result is that r ∈ R gives rise to a symmetric shift radix system τr if and only if r is contained in the union of three convex polyhedra (together with some parts of their boundary). We describe this set explicitly.

Cite this paper

@inproceedings{Huszti2007ThreedimensionalSS, title={Three-dimensional Symmetric Shift Radix Systems}, author={Andrea Huszti and Klaus Scheicher and P. Surer and J{\"{o}rg M. Thuswaldner}, year={2007} }