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Approximation measures for logarithms of algebraic numbers. (English) Zbl 1008.11028
Given a number field K and a number $$\xi$$ not in K, we say that $$\mu>0$$ is a K-irrationality measure of $$\xi$$ if, for any $$\varepsilon>0$$, we have $$\log|\xi-\beta|>-(1+\varepsilon)\mu h(\beta)$$ for all $$\beta$$ in K with sufficiently large logarithmic height $$h(\beta)$$. The authors seek K-irrationality measures for $$\log\alpha$$ with $$\alpha$$ in K. The starting point is the Padé approximation for $$\log z$$ which has the integral representation $a_n(z)+b_n(z)=I_n(z)=(z-1)^{2n+1}\int^1_0 \left({x(1-x)\over 1+x(z-1)}\right)^n{dx\over 1+x(z-1)}.$ The integral can be estimated by the saddle point method. Similarly, the polynomial $$b_n(z)$$ has an integral representation which can be treated by the saddle point method. Finally, $$h(a_n(\alpha)/b_n(\alpha))$$ can be estimated by elementary $$p$$-adic methods applied to the numerator and denominator of the fraction. This leads to the $${\mathbb{Q}}(\alpha)$$-irrationality measure of $$\log\alpha$$. Improved measures can be obtained by replacing the kernel $${(t-1)(z-t)\over t}$$ by $${(t-1)^j(z-t)^j\over t^{j-l}}$$. The integral now is a hypergeometric function, but again can be estimated by the saddle point method. Some examples of the results are: $\left|\log 2-{a+b\sqrt 2\over c+d\sqrt 2}\right|>C\max\{|a|,|b|,|c|,|d|\}^{-12.4288}$ for all integers $$a,b,c,d$$ with $$(c,d)\neq(0,0)$$ and, similarly, $\left|\pi-{a+b\sqrt 3\over c+d\sqrt 3}\right|>C\max\{|a|,|b|,|c|,|d|\}^{-46.9075}$ (based on $$\alpha=e^{\pi i/6}$$). The analysis is intricate and ingenious, but the introduction signposts the path in an admirably clear way.

##### MSC:
 11J82 Measures of irrationality and of transcendence 11J17 Approximation by numbers from a fixed field 30E15 Asymptotic representations in the complex plane
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