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Diophantine approximation in commutative algebraic groups. I: An effective version of the algebraic subgroup. (Approximation diophantienne dans les groupes algébriques commutatifs. I: Une version effective du sous-groupe algébrique.) (French) Zbl 0880.11054
The aim of this paper is to produce a general estimate of diophantine approximation. The main statement (which is too long to be explicitly stated here) includes several effective transcendence results for numbers related with the exponential map of a commutative algebraic group which is defined over the field of complex algebraic numbers.
Applications (which are not considered in the paper under review) include transcendence measures, lower bounds for linear combinations of logarithms or rational points on algebraic groups, density statements and distribution measure of rational points on an algebraic group (for instance on an abelian variety), as well as results of algebraic independence arising from measures of simultaneous approximation. Concerning this last topic, see the two following papers by D. Roy and M. Waldschmidt [Approximation diophantienne et indépendance algébrique de logarithmes, Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, 753-796 (1997)] and [Simultaneous approximation and algebraic independence, Ramanujan J. 4, 379-430 (1997)].

##### MSC:
 11J81 Transcendence (general theory) 11G10 Abelian varieties of dimension $$> 1$$ 14G05 Rational points 14L10 Group varieties
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