A class of maximally singular sets for rational approximation

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.