Geodesic
On Schmidt and Summerer parametric geometry of numbers
Damien Roy1
1 University of Ottawa, Ottawa, Ontario, Canada
Annals of mathematics, Tome 182 (2015) no. 2, pp. 739-786.

Voir la notice de l'article provenant de la source Annals of Mathematics website

Recently, W. M. Schmidt and L. Summerer introduced a new theory that allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in $\mathbb{R}^n$ and to discover new ones. They first note that these exponents can be computed in terms of the successive minima of a parametric family of convex bodies attached to the given point. Then they prove that the $n$-tuple of these successive minima can in turn be approximated up to bounded difference by a function from a certain class. In this paper, we show that the same is true within a smaller and simpler class of functions, which we call rigid systems. We also show that conversely, given a rigid system, there exists a point in $\mathbb{R}^n$ whose associated family of convex bodies has successive minima that approximate that rigid system up to bounded difference. As a consequence, the problem of describing the joint spectrum of a family of exponents of Diophantine approximation is reduced to combinatorial analysis.
DOI : 10.4007/annals.2015.182.2.9

Damien Roy 1

1 University of Ottawa, Ottawa, Ontario, Canada 
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Damien Roy. On Schmidt and Summerer parametric geometry of numbers. Annals of mathematics, Tome 182 (2015) no. 2, pp. 739-786. doi : 10.4007/annals.2015.182.2.9. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.182.2.9/

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