An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation

TL;DR

This paper establishes an optimal lower bound for the ratio between ordinary and uniform exponents in Diophantine approximation across any dimension, confirming a conjecture by Schmidt and Summerer.

Contribution

It proves the conjectured lower bound for all dimensions using regular systems in parametric geometry of numbers, extending previous results from dimensions 2 and 3.

Findings

Lower bound for the ratio is proven to be optimal.

The bound is achieved by regular systems in parametric geometry of numbers.

The result generalizes previous dimension-specific cases.

Abstract

We provide a lower bound for the ratio between the ordinary and uniform exponent of both simultaneous Diophantine approximation and Diophantine approximation by linear forms in any dimension. This lower bound was conjectured by Schmidt and Summerer and already shown in dimension $2$ and $3$. This lower bound is reached at regular systems presented in the context of parametric geometry of numbers, and thus optimal.