Hankel Determinants, Pade Approximations, and Irrationality Exponents

Yann BUGEAUD I.R.M.A., Universite de Strasbourg et CNRS Guo-Niu HAN I.R.M.A., Universite de Strasbourg et CNRS Zhi-Ying WEN Department of Mathematics, Tsinghua University Jia-Yan YAO Department of Mathematics, Tsinghua University

Number Theory mathscidoc:1611.24012

Int. Math. Res. Not. IMRN , 2016, (5), 1467–1496, 2016.6
The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.
Hankel determinant, continued fraction, automatic sequence, Thue--Morse sequence, periodicity, regular paperfolding sequence, Stern sequence, irrationality exponent
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@inproceedings{yann2016hankel,
  title={      Hankel Determinants, Pade Approximations, and Irrationality Exponents},
  author={Yann BUGEAUD, Guo-Niu HAN, Zhi-Ying WEN, and Jia-Yan YAO},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161124162300956776639},
  booktitle={ Int. Math. Res. Not. IMRN },
  volume={2016},
  number={5},
  pages={1467–1496},
  year={2016},
}
Yann BUGEAUD, Guo-Niu HAN, Zhi-Ying WEN, and Jia-Yan YAO. Hankel Determinants, Pade Approximations, and Irrationality Exponents. 2016. Vol. 2016. In Int. Math. Res. Not. IMRN . pp.1467–1496. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161124162300956776639.
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