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We study periodic expansions in positional number systems with a base $$\beta \in \mathbb {C},\ |\beta |>1$$ β ∈ C , | β | > 1 , and with coefficients in a finite set of digits $$\mathcal {A}\subset \mathbb {C}.$$ A ⊂ C . We are interested in determining those algebraic bases for which there exists $$\mathcal {A}\subset \mathbb {Q}(\beta ),$$ A ⊂ Q ( β ) , such that all elements of $$\mathbb {Q}(\beta )$$ Q ( β ) admit at least one eventually periodic representation with digits in $$\mathcal {A}$$ A . In this paper we prove a general result that guarantees the existence of such an $$\mathcal {A}$$ A . This result implies the existence of such an $$\mathcal {A}$$ A when $$\beta $$ β is a rational number or an algebraic integer with no conjugates of modulus 1. We also consider periodic representations of elements of $$\mathbb {Q}(\beta )$$ Q ( β ) for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of $$\mathbb {Q}(\beta )$$ Q ( β ) admits such a representation then $$\beta $$ β must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt (Bull Lond Math Soc 12(4):269–278, 1980).
Monatshefte f�r Mathematik – Springer Journals
Published: May 29, 2017
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