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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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##### References:
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