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.