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

Antoine Marnat; Nikolay Moshchevitin

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

Antoine Marnat, Nikolay Moshchevitin

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article

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.

Original language	Undefined/Unknown
Journal	arXiv.org e-Print archive
Publication status	Published - 8 Feb 2018

Keywords

math.NT

Cite this

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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.",

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TY - JOUR

T1 - An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation

AU - Marnat, Antoine

AU - Moshchevitin, Nikolay

PY - 2018/2/8

Y1 - 2018/2/8

N2 - 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.

AB - 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.