A holonomy reduction to equiangular lines

Definitions and mathematical statement

Let $1\le k\le d$ and $0\le\alpha\le1$. For a linear subspace $V\subseteq\mathbb R^d$, let $P_V$ be orthogonal projection onto $V$. Two $k$-dimensional subspaces $U,V$ are isoclinic with parameter $\alpha$ if $\|P_Vu\|=\alpha$ for every unit vector $u\in U$. Equivalently, \[ P_UP_VP_U=\alpha^2P_U. \] For equal-dimensional subspaces this condition is symmetric. The parameter used in the workshop is a projection length; the corresponding principal angle is $\arccos\alpha$. Let $N_\alpha^k(d)$ be the largest cardinality of a family of distinct $k$-dimensional subspaces of $\mathbb R^d$ in which every pair satisfies this condition with the same $\alpha$.

The two proposed bounds are:

1. For every fixed $k$ and $\alpha$, there is a finite constant $C_{k,\alpha}$ such that $N_\alpha^k(d)\le C_{k,\alpha}d$ for all $d\ge k$.
2. For every fixed $k$, there is a constant $C_k$ such that for each fixed $\alpha$, $\limsup_{d\to\infty}N_\alpha^k(d)/d\le C_k$.

The second assertion expresses $O_k(d)+o_{\alpha,k}(d)$: its leading coefficient is independent of $\alpha$, although the dimension from which an approximate bound holds may depend on $\alpha$. The substantive range is $0<\alpha<1$; at $\alpha=0$ the subspaces are mutually orthogonal, and at $\alpha=1$ two distinct subspaces cannot satisfy the condition.

Short English statement

Must an equi-isoclinic family of fixed-dimensional real subspaces grow only linearly with the ambient dimension, with a leading constant independent of the common angle?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000889 (AIM-COMBINATORICS-0014), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, High-dimensional phenomena in discrete analysis, item 4.1. Original workshop question underlying AIM-COMBINATORICS-0014. http://aimpl.org/highdimdiscrete/4/.