Simons Collaboration on Algorithms and Geometry Monthly Meeting, April 2021

Every measure on the sphere has a sub-Gaussian orthonormal basis

Speakers:
Ashwin Sah, Mehtaab Sawhney, Yufei Zhao

Does every Cayley graph have an eigenbasis with certain desirable boundedness properties? This problem was initially motivated by applications to graph expansion, and it led us to interesting problems (some still open) in high dimensional geometry.

We show that every probability measure on a high dimensional unit ball can be rotated so that all coordinate marginals are sub-Gaussian. This implies the existence of an \(L^p\)-bounded eigenbasis for vertex-transitive graphs. We also construct Cayley graphs where such bounds are nearly optimal.

(based on joint work with Assaf Naor)