A graph arising in the Geometry of Numbers

TL;DR

This paper extends the concept of the regular graph from the parametric geometry of numbers to the weighted case, providing a new framework for understanding extremal simultaneous approximation properties.

Contribution

It introduces and constructs an analogue of the regular graph for weighted simultaneous approximation, advancing the geometric understanding of approximation exponents.

Findings

Defined a weighted analogue of the regular graph.

Constructed the weighted regular graph explicitly.

Provided insights into extremal approximation inequalities.

Abstract

The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among classical exponents of simultaneous approximation can be guessed by a study of these graphs; in particular the so called regular graph is of major importance as it provides an extremal case for some of these inequalities. The aim of this paper is to define and construct an analogue of the regular graph in the case of weighted simultaneous approximation.