Given a positive integer M and a real number q∈(1,M+1], an expansion of a real number x∈[0,M/(q−1)] over the alphabet A={0,1,…,M} is a sequence (ci)∈AN such that x=∑i=1∞ciq−i. Generalizing many earlier results, we investigate in this paper the topological properties of the set Uq consisting of numbers x having a unique expansion of this form, and the combinatorial properties of the set Uq′ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [3] by adapting the method given in [12] for the case M=1.
Topology of univoque sets in real base expansions / de Vries, M.; Komornik, V.; Loreti, P.. - In: TOPOLOGY AND ITS APPLICATIONS. - ISSN 0166-8641. - 312:(2022), p. 108085. [10.1016/j.topol.2022.108085]
Topology of univoque sets in real base expansions
Loreti P.
2022
Abstract
Given a positive integer M and a real number q∈(1,M+1], an expansion of a real number x∈[0,M/(q−1)] over the alphabet A={0,1,…,M} is a sequence (ci)∈AN such that x=∑i=1∞ciq−i. Generalizing many earlier results, we investigate in this paper the topological properties of the set Uq consisting of numbers x having a unique expansion of this form, and the combinatorial properties of the set Uq′ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [3] by adapting the method given in [12] for the case M=1.| File | Dimensione | Formato | |
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