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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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