On the irrationality measure of \(\log 3\). (English) Zbl 1303.11081
V. Kh. Salikhov [Dokl. Math. 76, No. 3, 955–957 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 417, No. 6, 753–755 (2007; Zbl 1169.11032)] proved that the irrationality measure of \(\log 3\) has an upper bound \(\mu(\log 3)<5.126\). In the present paper the authors improved Salikhov’s method and found a better upper bound \(\mu(\log 3)<5.117\).
This bound follows from a more general result stating that \(\left| p+q_1\log 2+q_2\log 3\right|>H^{-4.117}\) for every \(p,q_1,q_2\) with sufficiently large \(H=\max\{|q_1|,|q_2|\}\).
The authors used semi-infinite linear programming and the LLL algorithm to find suitable polynomials appearing in the proof.
This bound follows from a more general result stating that \(\left| p+q_1\log 2+q_2\log 3\right|>H^{-4.117}\) for every \(p,q_1,q_2\) with sufficiently large \(H=\max\{|q_1|,|q_2|\}\).
The authors used semi-infinite linear programming and the LLL algorithm to find suitable polynomials appearing in the proof.
Reviewer: Jan Šustek (Ostrava)
MSC:
| 11J82 | Measures of irrationality and of transcendence |
| 11J86 | Linear forms in logarithms; Baker’s method |
Keywords:
irrationality measure; linear independence measure; integer transfinite diameter; semi-infinite linear programming; LLL algorithmCitations:
Zbl 1169.11032References:
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