##
**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.

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.

Reviewer: Simon Kristensen (Aarhus)

Full Text:
DOI

### References:

[1] | Adamczewski, B., Bugeaud, Y.: Mesures de transcendance et aspects quantitatifs de la méthode de Thue–Siegel–Roth–Schmidt. Preprint · Zbl 1200.11054 |

[2] | Allouche J.-P. and Shallit J.O. (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge · Zbl 1086.11015 |

[3] | Beresnevich V., Dickinson H. and Velani S.L. (2001). Sets of exact logarithmic order in the theory of diophantine approximation. Math. Ann. 321: 253–273 · Zbl 1006.11039 |

[4] | Bugeaud Y. (2003). Sets of exact approximation order by rational numbers. Math. Ann. 327: 171–190 · Zbl 1044.11059 |

[5] | Bugeaud Y. (2004). Approximation by algebraic numbers. Cambridge Tracts in Mathematics 160, Cambridge · Zbl 1055.11002 |

[6] | Jarní k V. (1931). Über die simultanen diophantische approximationen. Math. Z. 33: 505–543 · JFM 57.1370.01 |

[7] | Kleinbock D., Lindenstrauss E. and Weiss B. (2004). On fractal measures and diophantine approximation. Select. Math. 10: 479–523 · Zbl 1130.11039 |

[8] | Kleinbock D. and Weiss B. (2005). Badly approximable vectors on fractals. Isr. J. Math. 149: 137–170 · Zbl 1092.28004 |

[9] | Kristensen S. (2006). Approximating numbers with missing digits by algebraic numbers. Proc. Edinb. Math. Soc. 49: 657–666 · Zbl 1161.11024 |

[10] | Kristensen S., Thorn R. and Velani S. (2006). Diophantine approximation and badly approximable sets. Adv. Math. 203: 132–169 · Zbl 1098.11039 |

[11] | Levesley J., Salp C. and Velani S.L. (2007). On a problem of K Mahler: diophantine approximation and cantor sets. Math. Ann. 338: 97–118 · Zbl 1115.11040 |

[12] | Mahler K. (1984). Some suggestions for further research. Bull. Austral. Math. Soc. 29: 101–108 · Zbl 0517.10001 |

[13] | van der Poorten A.J. and Shallit J. (1992). Folded continued fractions. J. Number Theory 40: 237–250 · Zbl 0753.11005 |

[14] | Ridout D. (1957). Rational approximations to algebraic numbers. Mathematika 4: 125–131 · Zbl 0079.27401 |

[15] | Schmidt, W.M.: Approximation to algebraic numbers. Monographie de l’Enseignement Mathématique 19, Genève, (1971) |

[16] | Shallit J. (1979). Simple continued fractions for some irrational numbers. J. Number Theory 11: 209–217 · Zbl 0404.10003 |

[17] | Weiss B. (2001). Almost no points on a Cantor set are very well approximable. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457: 949–952 · Zbl 0997.11063 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.