A gap of the exponents of repetitions of Sturmian words

By measuring second occurring times of factors of an infinite word $x$ , Bugeaud and Kim introduced a new quantity $rep (x)$ called the exponent of repetition of $x$ . It was proved by Bugeaud and Kim that $1 \leq rep (x) \leq r_{max} = \sqrt{10} - \frac{3}{2}$ if $x$ is a Sturmian word. We determine the value $r_{1}$ such that there is no Sturmian word $x$ satisfying $r_{1} < rep (x) < r_{max}$ and $r_{1}$ is an accumulation point of the set of $rep (x)$ when $x$ runs over the Sturmian words.

To the memory of Professor Ichiro Satake (1927–2014)

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Keywords

combinatorics on words, Sturmian word, continued fraction, irrationality exponent, irrationality measure

Mathematical Subject Classification

Primary: 68R15

Secondary: 11A55, 11A63

Milestones

Received: 9 January 2021

Revised: 18 May 2021

Accepted: 7 June 2021

Published: 13 September 2021

Authors

Suzue Ohnaka
Naruo Senior High School Nishinomiya Japan

Takao Watanabe
Department of Mathematics Graduate School of Science Osaka University Toyonaka Japan

Vol. 10, No. 3, 2021

Suzue Ohnaka and Takao Watanabe

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors