Acta Mathematica Hungarica

, Volume 153, Issue 2, pp 318–333

# Finiteness in real real cubic fields

• Z. Masáková
• M. Tinková
Article

## Abstract

We study finiteness property in numeration systems with cubic Pisot unit base. A base β > 1 is said to satisfy property (F), if the set Fin (β) of numbers with finite β-expansions forms a ring. We show that in every real cubic field which is not totally real, there exists a cubic Pisot unit satisfying (F). On the other hand, there exist totally real cubic fields without such a unit. In such fields, however, one finds a cubic Pisot unit β > 1 satisfying property (−F), i.e., the set Fin (−β) of finite (−β)-expansions forms a ring.

## Key words and phrases

beta-expansion finiteness Pisot unit cubic field

11A63 11R16

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