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For θ a non-algebraic point on a quasi projective variety over a number field, I prove that θ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result include a proof of a slightly strengthend version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theorey on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields. Comments : There will be two more papers by the author in this series as well as several papers on applications in algebraic independence theory. |