Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer

TL;DR

This paper demonstrates that for regular systems, it is possible to construct points in al R^n whose associated convex bodies' successive minima approximate given functions, advancing the theory of Diophantine approximation exponents.

Contribution

It proves the existence of points corresponding to regular systems, establishing a key converse in the framework of Schmidt and Summerer's theory.

Findings

Existence of points for regular systems with prescribed successive minima

Advancement in understanding the link between convex bodies and Diophantine exponents

Extension of the theory to a broader class of functions

Abstract

In a series of recent papers, W. M. Schmidt and L. Summerer developed a new theory by which they recover all major generic inequalities relating exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and find new ones. Given a point in $\mathbb{R}^n$, they first show how most of its exponents of Diophantine approximation can be computed in terms of the successive minima of a parametric family of convex bodies attached to that point. Then they prove that these successive minima can in turn be approximated by a certain class of functions which they call $(n,\gamma)$-systems. In this way, they bring the whole problem to the study of these functions. To complete the theory, one would like to know if, conversely, given an $(n,\gamma)$-system, there exists a point in $\mathbb{R}^n$ whose associated family of convex bodies has successive minima which approximate that function. In the present paper, we show that this is true for a class of functions which they call regular systems.