On Schmidt and Summerer parametric geometry of numbers

Abstract

Recently, W. M. Schmidt and L. Summerer introduced a new theory which 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 which 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.

40 pages, 3 figures

References in corpus (1)

• On transfer inequalities in Diophantine approximation

Cited by in corpus (21)

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• A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation
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• Generalizations of a result of Jarnik on simultaneous approximation
• Rational approximation to real points on quadratic hypersurfaces
• Diophantine sets and Dirichlet improvability
• Padé approximation for a class of hypergeometric functions and parametric geometry of numbers
• Going-up theorems for simultaneous Diophantine approximation
• Exponents of Diophantine approximation in dimension two for a general class of numbers
• Geometry of Diophantine exponents
• Divergent trajectories on products of homogeneous spaces
• On a Conjecture of Schmidt for the Parametric Geometry of Numbers
• Hausdorff and packing dimension of Diophantine sets
• Parametric Geometry of Numbers with General Flow
• K-divergent lattices
• On Approximation constants for Liouville numbers
• On badly approximable numbers
• On connectedness in the parametric geometry of numbers
• Construction of numbers with almost all convergents in a Cantor set