An inequality for Diophantine exponents in the case of simultaneous approximation

TL;DR

This paper establishes a new inequality relating individual and uniform Diophantine exponents for simultaneous approximation, improving upon Jarnik's inequality for small uniform exponents.

Contribution

The paper introduces a novel inequality for Diophantine exponents that outperforms Jarnik's inequality in certain cases of simultaneous approximation.

Findings

New inequality for Diophantine exponents

Improves upon Jarnik's inequality for small uniform exponents

Advances understanding of simultaneous Diophantine approximation

Abstract

In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.