Singular Vectors in Real Affine Subspaces

TL;DR

This paper investigates the measure zero properties of singular vectors in affine subspaces, linking these properties to the rationality and singularity of the parametrizing matrices, with implications for Diophantine approximation.

Contribution

It establishes a connection between measure zero sets of singular vectors and the non-$n$-singularity of matrices parametrizing affine subspaces, including hyperplanes.

Findings

Singular vectors in certain affine subspaces have measure zero under specific conditions.

The measure zero property is characterized by the non-$n$-singularity of the parametrizing matrix.

For affine hyperplanes, measure zero of singular vectors occurs if and only if the matrix is not rational.

Abstract

We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside those affine subspaces. We define a notion of $n$-singularity for matrices, which is closely related to the uniform exponent of irrationality. For certain affine subspaces, we show that the set of singular vectors has measure zero if and only if the parametrizing matrix is not $n$-singular. In particular, we show for affine hyperplanes the set of singular vectors has measure zero if and only if the parametrizing matrix is not rational.