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Zbl 1008.11028
Amoroso, Francesco; Viola, Carlo
Approximation measures for logarithms of algebraic numbers.
(English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No.1, 225-249 (2001). ISSN 0391-173X

Given a number field {\bf K} and a number $\xi$ not in {\bf K}, we say that $\mu>0$ is a {\bf K}-irrationality measure of $\xi$ if, for any $\varepsilon>0$, we have $\log|\xi-\beta|>-(1+\varepsilon)\mu h(\beta)$ for all $\beta$ in {\bf K} with sufficiently large logarithmic height $h(\beta)$. The authors seek {\bf K}-irrationality measures for $\log\alpha$ with $\alpha$ in {\bf K}. The starting point is the PadÃ© 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 ${\bbfQ}(\alpha)$-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 $(c,d)\ne(0,0)$ 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.
[John Loxton (North Ryde)]
MSC 2000:
*11J82 Measures of irrationality and of transcendence
11J17 Approximation by numbers from a fixed field
30E15 Asymptotic representations in the complex domain

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