[en] For an alternate base B=(β_0,...,β_{p-1}), we show that if all rational numbers in the unit interval [0, 1) have periodic expansions with respect to all shifts of B, then the bases β_0,...,β_{p-1} all belong to the extension field Q(β) where β is the product
β_0 · · · β_{p−1} and moreover, this product β must be either a Pisot number or a Salem number. We also prove the stronger statement that if the bases β_0, ..., β_{p−1} belong to Q(β) but the product β is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic β-expansion is nowhere dense in [0, 1). Moreover, in the case where the product β is a Pisot number and the bases β_0, ..., β_{p−1} all belong to Q(β), we prove that the set of points in [0, 1) having an ultimately periodic β-expansion is precisely the set Q(β) ∩ [0, 1).
Disciplines :
Mathematics
Author, co-author :
Charlier, Emilie ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Cisternino, Célia ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Kreczman, Savinien ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Language :
English
Title :
On periodic alternate base expansions
Publication date :
2022
Event name :
Journées montoises d'informatique théorique
Event place :
Prague, Czechia
Event date :
du 5 septembre 2022 au 9 septembre 2022
Audience :
International
Main work title :
Actes des 18èmes Journées Montoises d'Informatique Théorique
Publisher :
Czech Technical University in Prague, Prague, Czechia
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