Hankel Determinants, Pade Approximations, and Irrationality Exponents
Hankel Determinants, Pade Approximations, and Irrationality Exponents
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Hankel 行列式、Pade 近似和无理数指数
DOI：
10.1093/imrn/rnv185
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发表时间：
2016
影响因子：
1
通讯作者：
Yao Jia-Yan
中科院分区：
文献类型：
--
作者：
Bugeaud Yann;Han Guo-Niu;Wen Zhi-Ying;Yao Jia-Yan
一个无理数的无理数指数，它衡量有理数对的逼近速度，一般来说是很难明确计算的，除非我们知道的连分式展开。迄今取得的成果相当零碎，往往是个案处理。在这篇文章中，我们将通过证明大类自动数和马勒数（超越的）的无理指数恰好等于2来统一关于这个问题的所有已知结果。我们的类包含Thue-Morse-Mahler数、Fermat数的倒数之和、正规折纸数，这些数以前分别由Bugeaud、Coons、Guo、Wu和Wen考虑过，但也有新的类，如Stern数等，在其他成分中，我们的证明使用了Hankel行列式的结果。
The irrationality exponent of an irrational number, which measures the approximation rate ofby rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of. 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.