Upper Bounds for $s$-Distance Subspaces

TL;DR

This work advances the study of s-distance subspaces in Gr$(k,n)$ by developing a refined polynomial-method framework that leverages both projective and affine Grassmannian embeddings. By combining Hilbert-function bounds with a refined dimension-counting technique, the authors derive explicit upper bounds for N$_s^{d_C}$ and N$_s^{d_{FS}}$ that improve previous results and specify the leading terms in terms of $n$, $k$, and $s$. They also connect spherical/line subspace bounds to Grassmannian geometry, and provide new bounds for equiangular subspaces (angle distances), tightening the asymptotics and revealing unattainable target bounds in earlier formulations. The results contribute sharp asymptotic insights for extremal configurations in Grassmannians, with potential implications for coding theory and related geometric combinatorics.

Abstract

As a generalization of equiangular lines, equiangular subspaces were first systematically studied by Balla, Dräxler, Keevash and Sudakov in 2017. In this paper, we extend their work to $s$-distance subspaces, i.e., to sets of $k$-dimensional subspaces in $\mathbb{R}^n$ whose pairwise distances take $s$ distinct values. We establish upper bounds on the maximum cardinality of such sets. In particular, our bounds generalize and improve results of Balla and Sudakov.

Theorems & Definitions (20)

• Lemma 2.1
• Lemma 2.2: Hilbert polynomial of Grassmannian
• Proposition 3.1: Refined dimension counting
• proof
• Remark 3.2
• Theorem 4.1: $s$-distance subspaces for $d_{{C}}$
• proof
• Remark 4.2
• Lemma 4.3
• proof