Non-planarity and metric Diophantine approximation for systems of linear forms

TL;DR

This paper introduces a new concept of weak non-planarity for manifolds and measures, advancing the understanding of metric Diophantine approximation for systems of linear forms and extending inhomogeneous and weighted theories.

Contribution

It develops a general theory of metric Diophantine approximation using weak non-planarity, generalizes non-planarity, and extends inhomogeneous and weighted approximation results.

Findings

Established strong extremality of manifolds and measures.

Introduced the concept of weak non-planarity, shown to be near optimal.

Extended inhomogeneous transference results to balance with homogeneous theory.

Abstract

In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of $m\times n$ matrices over $\Bbb R$ is introduced and studied. This notion generalises the one of non-planarity in $\Bbb R^n$ and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.