Amoroso, Francesco; Viola, Carlo Approximation measures for logarithms of algebraic numbers. (English) Zbl 1008.11028 Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No. 1, 225-249 (2001). 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. Reviewer: John Loxton (North Ryde) Cited in 1 ReviewCited in 9 Documents MSC: 11J82 Measures of irrationality and of transcendence 11J17 Approximation by numbers from a fixed field 30E15 Asymptotic representations in the complex plane Keywords:irrationality measure; Padé approximation; saddle point method PDF BibTeX XML Cite \textit{F. Amoroso} and \textit{C. Viola}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No. 1, 225--249 (2001; Zbl 1008.11028) Full Text: Numdam EuDML OpenURL References: [1] K. Alladi - M.L. Robinson , Legendre polynomials and irrationality , J. reine angew. Math. 318 ( 1980 ), 137 - 155 . Article | MR 579389 | Zbl 0425.10039 · Zbl 0425.10039 [2] J. Dieudonné , ” Calcul Infinitésimal ”, Hermann , Paris , 1968 . MR 226971 | Zbl 0155.10001 · Zbl 0155.10001 [3] M. Hata , C2-saddle method and Beukers’ integral , Trans. Amer. Math. Soc. 352 ( 2000 ), 4557 - 4583 . MR 1641099 | Zbl 0996.11048 · Zbl 0996.11048 [4] E. Reyssat , Mesures de transcendance pour les logarithmes de nombres rationnels , In: ” Approximations diophantiennes et nombres transcendants ”, D. Bertrand - M. Waldschmidt (eds.), Progress in Math . 31 , Birkhäuser , 1983 , pp. 235 - 245 . MR 702201 | Zbl 0522.10023 · Zbl 0522.10023 [5] G. Rhin - C. Viola , On a permutation group related to \zeta (2) , Acta Arith. 77 ( 1996 ), 23 - 56 . Article | Zbl 0864.11037 · Zbl 0864.11037 [6] W.M. Schmidt , ” Diophantine Approximation ”, Lecture Notes in Mathematics 785 , Springer-Verlag , 1980 . MR 568710 | Zbl 0421.10019 · Zbl 0421.10019 [7] C. Viola , Hypergeometric functions and irrationality measures , In: ” ’Analytic Number Theory ”, Y. Motohashi (ed.), London Math. Soc. Lecture Note Series 247 , Cambridge Univ. Press , 1997 , pp. 353 - 360 . MR 1695002 | Zbl 0904.11020 · Zbl 0904.11020 [8] M. Waldschmidt , ” Nombres Transcendants ”, Lecture Notes in Mathematics 402 , Springer-Verlag , 1974 . MR 360483 | Zbl 0302.10030 · Zbl 0302.10030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.