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Erschienen in:
2016 | OriginalPaper | Buchkapitel
A Geometric Face of Diophantine Analysis
verfasst von : Tapani Matala-aho
Erschienen in: Diophantine Analysis
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Abstract
Geometry of numbers is a powerful tool in studying Diophantine inequalities. In geometry of numbers a basic question is to find a non-zero lattice vector from a convex subset in an n-dimensional space, say in \(\mathbb {R}^n\). Hermann Minkowski answered this challenge with his convex body theorems. In these lectures we shall discuss how to apply Minkowski’s theorems to prove classical Diophantine inequalities and some variations of Siegel’s lemma. Further, we shall shortly discuss corresponding inequalities over imaginary quadratic fields. From the nature of the above results follows that a lower bound for the absolute value of an arbitrary non-zero linear form in m linearly independent numbers is not bigger than a certain positive function depending on the coefficients and the number of variables of the linear form. For a concrete set of numbers it is a big challenge to find such lower bounds. We will give a recent example on such lower bounds, namely a new generalised transcendence measure for e.