A class of maximally singular sets for rational approximation

TL;DR

This paper introduces a criterion for identifying maximally singular sets in projective space, including Grassmannians and certain quadratic hypersurfaces, based on their rational approximation properties.

Contribution

It provides a new criterion for maximal singularity in rational approximation, encompassing various geometric sets like Grassmannians and quadratic hypersurfaces.

Findings

Identifies many maximally singular sets including Grassmannians.

Recovers a known result about quadratic hypersurfaces.

Establishes a criterion linking geometric sets to rational approximation exponents.

Abstract

We say that a subset of $\mathbb{P}^n(\mathbb{R})$ is maximally singular if its contains points with $\mathbb{Q}$-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to $1$, the maximal possible value. In this paper, we give a criterion which provides many such sets including Grassmannians. We also recover a result of the author and Roy about a class of quadratic hypersurfaces.