On the expansions of real numbers in two integer bases

Yann Bugeaud; Dong Han Kim

doi:10.5802/aif.3134

On the expansions of real numbers in two integer bases

Yann Bugeaud
, Dong Han Kim

Department of Mathematics Education

Université de Strasbourg

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

Abstract

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0, 1, . . . , r-1} and {0, 1, . . . , s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

Original language	English
Pages (from-to)	2225-2235
Number of pages	11
Journal	Annales de l'Institut Fourier
Volume	67
Issue number	5
DOIs	https://doi.org/10.5802/aif.3134
State	Published - 2017

Keywords

Combinatorics on words
Complexity
Continued fraction
Integer base expansion
Sturmian word

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10.5802/aif.3134

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abstract = "Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on \{0, 1, . . . , r-1\} and \{0, 1, . . . , s-1\}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.",

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Y1 - 2017

N2 - Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0, 1, . . . , r-1} and {0, 1, . . . , s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

AB - Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0, 1, . . . , r-1} and {0, 1, . . . , s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

KW - Combinatorics on words

KW - Complexity

KW - Continued fraction

KW - Integer base expansion

KW - Sturmian word

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JO - Annales de l'Institut Fourier

JF - Annales de l'Institut Fourier

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ER -

On the expansions of real numbers in two integer bases

Abstract

Keywords

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