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A geometric face of Diophantine analysis. (English) Zbl 1416.11121
Steuding, Jörn (ed.), Diophantine analysis. Course notes from a summer school, Würzburg, Germany, July 21–24, 2014. Basel: Birkhäuser/Springer. Trends Math., 129-174 (2016).
Summary: 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 \(e\).
For the entire collection see [Zbl 1364.11007].

11H06 Lattices and convex bodies (number-theoretic aspects)
11J25 Diophantine inequalities
11J81 Transcendence (general theory)
11J82 Measures of irrationality and of transcendence
Full Text: DOI
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