Vol. 10, No. 3, 2021

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A gap of the exponents of repetitions of Sturmian words

Suzue Ohnaka and Takao Watanabe

Vol. 10 (2021), No. 3, 203–234

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 rep(x) rmax = 10 3 2 if x is a Sturmian word. We determine the value r1 such that there is no Sturmian word x satisfying r1 < rep(x) < rmax and r1 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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combinatorics on words, Sturmian word, continued fraction, irrationality exponent, irrationality measure
Mathematical Subject Classification
Primary: 68R15
Secondary: 11A55, 11A63
Received: 9 January 2021
Revised: 18 May 2021
Accepted: 7 June 2021
Published: 13 September 2021
Suzue Ohnaka
Naruo Senior High School
Takao Watanabe
Department of Mathematics
Graduate School of Science
Osaka University