The cylindrical width of transitive sets

TL;DR

This paper extends Green's result by showing that finite transitive sets in high-dimensional space are close to low-codimension subspaces, with bounds that are proven to be optimal.

Contribution

It generalizes Green's theorem from codimension 1 to higher codimensions, establishing optimal bounds for the proximity of transitive sets to subspaces.

Findings

Finite transitive sets are within a specific distance of low-codimension subspaces.

The distance bounds are proven to be optimal.

The result applies for a range of codimensions up to d/(log d)^C.

Abstract

We show that for every $1 \le k \le d/(\log d)^C$, every finite transitive set of unit vectors in $\mathbb{R}^d$ lies within distance $O(1/\sqrt{\log (d/k)})$ of some codimension $k$ subspace, and this distance bound is best possible. This extends a result of Ben Green, who proved it for $k=1$.