Some notes on the regular graph defined by Schmidt and Summerer and uniform approximation

TL;DR

This paper investigates the properties of regular graphs in the parametric geometry of numbers, focusing on the behavior of successive minima functions and improving bounds on approximation constants under certain conjectures.

Contribution

It provides new results on successive minima functions for regular graphs and refines upper bounds for approximation constants assuming the Schmidt-Summerer conjecture.

Findings

New results on the behavior of successive minima functions for regular graphs.

Improved upper bounds for the approximation constants w_n() for all transcendental .

Results depend on the validity of the Schmidt-Summerer conjecture.

Abstract

Within the study of parametric geometry of numbers W. Schmidt and L. Summerer introduced so-called regular graphs. Roughly speaking the successive minima functions for the classical simultaneous Diophantine approximation problem have a very special pattern if the vector $\underline{\zeta}$ induces a regular graph. The regular graph is in particular of interest due to a conjecture by Schmidt and Summer concerning classic approximation constants. This paper aims to provide several new results on the behavior of the successive minima functions for the regular graph. Moreover, we improve the best known upper bounds for the classic approximation constants $\hat{w}_{n}(\zeta)$, valid uniformly for all transcendental $\zeta$, provided that the Schmidt-Summerer conjecture is true.