Vol. 315, No. 1, 2021

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On the Lévy constants of Sturmian continued fractions

Yann Bugeaud, Dong Han Kim and Seul Bee Lee

Vol. 315 (2021), No. 1, 1–25

The Lévy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a Lévy constant. Let a,b be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over {a,b}, then it has a Lévy constant, which depends only on ab, and the slope of the Sturmian sequence, but not on its intercept. We show that the set of Lévy constants of irrational real numbers whose sequence of partial quotients is periodic or Sturmian is equal to the whole interval [log((1 + 5)2),+).

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continued fraction, Lévy constant, Sturmian word, mechanical word, quasi-Sturmian word
Mathematical Subject Classification
Primary: 11A55
Secondary: 68R15
Received: 20 April 2020
Revised: 23 July 2021
Accepted: 17 September 2021
Published: 13 December 2021
Yann Bugeaud
Université de Strasbourg
Dong Han Kim
Dongguk University
South Korea
Seul Bee Lee
Scuola Normale Superiore