On the Lévy constants of Sturmian continued fractions

Yann Bugeaud, Dong Han Kim and Seul Bee Lee

Vol. 315 (2021), No. 1, 1–25

DOI: 10.2140/pjm.2021.315.1

Abstract

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 $\left\{a,b\right\}$, then it has a Lévy constant, which depends only on $a$,  $b$, 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 $\left[log\left(\left(1+\sqrt{5}\right)∕2\right),+\infty \right)$.

Keywords

continued fraction, Lévy constant, Sturmian word, mechanical word, quasi-Sturmian word

Mathematical Subject Classification

Primary: 11A55

Secondary: 68R15

Milestones

Received: 20 April 2020

Revised: 23 July 2021

Accepted: 17 September 2021

Published: 13 December 2021