Independence of the Diophantine exponents associated with linear subspaces

TL;DR

This paper investigates the independence of Diophantine exponents related to linear subspaces in ^n, extending classical approximation theory and demonstrating the absence of smooth relations among certain associated functions.

Contribution

It introduces a novel analysis of Diophantine exponents for subspaces, proving their independence and constructing specific subspaces with computable exponents.

Findings

No smooth relations among functions associated with these exponents

Constructed subspaces with explicitly computable exponents

Extended classical Diophantine approximation to higher-dimensional subspaces

Abstract

We elaborate on a problem raised by Schmidt in 1967 which generalizes the theory of classical Diophantine approximation to subspaces of $\R^n$. We consider Diophantine exponents for linear subspaces of $\R^n$ which generalize the irrationality measure for real numbers. We prove here that we have no smooth relations among some functions associated to these exponents. To establish this result, we construct subspaces for which we are able to compute the exponents.