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\(\mathbf{C}^{2}\)-saddle method and Beukers’ integral. (English) Zbl 0996.11048
By considering modifying Beukers’ double integral [F. Beukers, Bull. Lond. Math. Soc. 11, 268-272 (1979; Zbl 0421.10023)] \[ J(z)= \iint_{R_k} \frac{P(x)Q(y)} {1-xyz} dx dy, \] where \(R_k\) is the rectangular region \((1, (k+1)/k)\times (k/(k+1),1)\), \(k\in \mathbb{N}\) and \(P(x)\), \(Q(y)\) are certain polynomials with integral coefficients, the author gives new non-quadraticity measures for the values of the logarithm at specific rational points as follows: For any \(\varepsilon>0\) there is an effective positive constant \(H_0(\varepsilon)\) such that \[ \bigl|\log (1+ \tfrac{1}{k})-\xi\bigr|\geq H^{-\sigma (k,\mu)/ \tau(k,\mu)-1- \varepsilon} \] for any quadratic number \(\xi\) with the height \(H= H(\xi)\geq H_0(\varepsilon)\), where \(k\in \mathbb{N}\), \(\mu\in (0,3-2 \sqrt{2})\) satisfy \(\tau(k,\mu)> 0\), and \(\sigma(k, u)\), \(\tau(k,\mu)\) are only dependent on \(P\) and \(Q\) (their explicit expressions are omitted here).
In particular, from it we deduce that \(|\log 2-\xi|\geq H^{-25.0463}\) for any quadratic number \(\xi\) with \(H(\xi)\geq H_0\). This improves the earlier results (exponents) 287.819 and 105 obtained by H. Cohen [Sémin. Théorie Nombres, Univ. Grenoble I 1980-1981, Exp. No. 1, 47 p. (1981; Zbl 0479.10022)] and E. Reyssat [Approximations diophantiennes et nombres transcendants, Colloq. Luminy/Fr. 1982, Prog. Math. 31, 235-245 (1983; Zbl 0522.10023)], respectively. In order to establish the result mentioned above, the author gives the two-dimensional version of the saddle method (the \(\mathbb{C}^2\)-saddle method), and studies several basic properties on saddles of rational functions.

MSC:
11J82 Measures of irrationality and of transcendence
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