Diophantine approximation and Cantor sets. (English) Zbl 1163.11056

The irrationality exponent of a real number \(\xi\) is the supremum over all \(\mu\) for which the inequality \[ \left| \xi - {p \over q}\right| < {1 \over {q^\mu}} \] has infinitely many rational solutions \(p/q\). For any real irrational number \(\xi\) the irrationality exponent \(\mu(\xi)\) is at least \(2\). The present paper is concerned with the possible values of this exponent as \(\xi\) varies over the middle third Cantor set, i.e., the set of real numbers that can be written in base \(3\) without the use of the digit \(1\).
It was shown by J. Levesley, C. Salp and S. L. Velani [Math. Ann. 338, No. 1, 97–118 (2007; Zbl 1115.11040)] that the middle third Cantor set contains elements with any prescribed irrationality exponent greater than or equal to \((3 + \sqrt{5})/2\). The present paper fills the gap by providing an explicit construction of uncountably many irrational numbers in the middle third Cantor set of any prescribed irrationality exponent greater than or equal to \(2\).
The ingenious proof depends on a Folding Lemma due to A. J. van der Poorten and J. Shallit [J. Number Theory 40, No. 2, 237–250 (1992; Zbl 0753.11005)]. It also applies to more general missing digit sets, and can be modified to prove the existence of automatic numbers with any prescribed rational irrationality exponent. In addition to the results of the paper, several interesting suggestions for further research are included.


11J82 Measures of irrationality and of transcendence
28A80 Fractals
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