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Algebraic numbers close to 1: Results and methods. (English) Zbl 0892.11023

Murty, V. Kumar (ed.) et al., Number theory. Proceedings of the international conference on discrete mathematics and number theory, Tiruchirapalli, India, January 3–6, 1996 on the occasion of the 10th anniversary of the Ramanujan Mathematical Society. Providence, RI: American Mathematical Society. Contemp. Math. 210, 305-316 (1998).
For algebraic \(\alpha\), lower bounds for \(|\alpha -1|\) by means of the degree and of Mahler’s measure of \(\alpha\) have been given by M. Mignotte [Ann. Fac. Sci. Toulouse, V. Sér., Math. 1, 165-170 (1979; Zbl 0421.10022)], M. Mignotte and M. Waldschmidt [J. Number Theory 47, 43-62 (1994; Zbl 0801.11033)], and Y. Bugeaud, M. Mignotte and F. Normandin [C. R. Acad. Sci., Paris, Sér. I 321, 517-522 (1995; Zbl 0848.11047)]. Here, the author describes and investigates the crucial steps in the proofs of these lower bounds. He also shows that the lower bound given by Mignotte and Waldschmidt is almost sharp, and finally discusses a generalization in several variables.
For the entire collection see [Zbl 0878.00049].

MSC:

11J68 Approximation to algebraic numbers
11J85 Algebraic independence; Gel’fond’s method
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