Abstract
We study periodic expansions in positional number systems with a base beta in mathbb {C}, |beta |>1, and with coefficients in a finite set of digits mathcal {A}subset mathbb {C}. We are interested in determining those algebraic bases for which there exists mathcal {A}subset mathbb {Q}(beta ), such that all elements of mathbb {Q}(beta ) admit at least one eventually periodic representation with digits in mathcal {A}. In this paper we prove a general result that guarantees the existence of such an mathcal {A}. This result implies the existence of such an mathcal {A} when beta 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 ) 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 ) admits such a representation then beta 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).
Highlights
We consider representations of numbers in a base β ∈ C, |β| > 1, using a finite alphabet of digits
We study periodic expansions in positional number systems with a base β ∈ C, |β| > 1, and with coefficients in a finite set of digits A ⊂ C
We prove that if every element of Q(β) admits such a representation β must be a Pisot number or a Salem number
Summary
We consider representations of numbers in a base β ∈ C, |β| > 1, using a finite alphabet of digits. Alphabets exist which yield eventually periodic representations for every element of Q(β) for very general bases, perhaps all algebraic β, Pisot and Salem numbers still play a crucial role in the study of our question. They are the only bases for which eventually periodic representations can satisfy a special property. The represented number is in its modulus proportional to the highest power of the base used, and the proportionality is uniform for every element of the field Q(β) We call such representations ‘weak greedy’, these ideas are expressed formally in Definition 13. This property of Pisot and Salem numbers may be seen as a strengthening of Schmidt’s theorem; we state it as Theorem 18
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