On Schmidt and Summerer parametric geometry of numbers

TL;DR

This paper refines Schmidt and Summerer's parametric geometry of numbers by demonstrating that rigid systems can effectively approximate successive minima, simplifying the analysis of Diophantine exponents.

Contribution

It introduces the concept of rigid systems as a simpler class of functions to approximate successive minima, linking geometric and combinatorial approaches in Diophantine approximation.

Findings

Rigid systems can approximate successive minima up to bounded difference.

Every rigid system corresponds to a point in n with similar properties.

The joint spectrum of Diophantine exponents reduces to combinatorial analysis.

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.