On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations

Yann Bugeaud; Dong Han Kim; Michel Laurent; Arnaldo Nogueira

doi:10.2422/2036-2145.202001_008

On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations

Yann Bugeaud
, Dong Han Kim
, Michel Laurent
, Arnaldo Nogueira

Department of Mathematics Education

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We prove that these Cantor sets are made up of transcendental numbers, up to their endpoints 0 and 1, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers 1; ∈1; ∈2, where ∈1 and ∈2 are two arbitrary Sturmian numbers with the same slope.

Original language	English
Pages (from-to)	1691-1704
Number of pages	14
Journal	Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Volume	22
Issue number	4
DOIs	https://doi.org/10.2422/2036-2145.202001_008
State	Published - 2021

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10.2422/2036-2145.202001_008

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abstract = "We prove that these Cantor sets are made up of transcendental numbers, up to their endpoints 0 and 1, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers 1; ∈1; ∈2, where ∈1 and ∈2 are two arbitrary Sturmian numbers with the same slope.",

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AU - Nogueira, Arnaldo

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AB - We prove that these Cantor sets are made up of transcendental numbers, up to their endpoints 0 and 1, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers 1; ∈1; ∈2, where ∈1 and ∈2 are two arbitrary Sturmian numbers with the same slope.

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On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations

Abstract

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