Singular Vectors in Real Affine Subspaces
Number Theory
2022-08-30 v2 Dynamical Systems
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 -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 -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.
Cite
@article{arxiv.2208.02212,
title = {Singular Vectors in Real Affine Subspaces},
author = {Shreyasi Datta and Yewei Xu},
journal= {arXiv preprint arXiv:2208.02212},
year = {2022}
}
Comments
11 pages. Theorem 1.2 is improved by fixing an error in a lemma. arXiv admin note: substantial text overlap with arXiv:2203.09716