Upper Bounds for $s$-Distance Subspaces

TL;DR

This paper extends the concept of equiangular lines to $s$-distance subspaces, providing new upper bounds on the maximum size of sets of subspaces with limited pairwise distances, generalizing previous results.

Contribution

It introduces upper bounds for the size of $s$-distance subspace sets, advancing the understanding of geometric configurations in high-dimensional spaces.

Findings

Established upper bounds on the cardinality of $s$-distance subspace sets

Generalized and improved previous bounds for equiangular lines

Extended the theory to sets with multiple distinct pairwise distances

Abstract

As a generalization of equiangular lines, equiangular subspaces were first systematically studied by Balla, Dr\"{a}xler, 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.