Detecting Fourier subspaces

TL;DR

This paper investigates the relationship between Fourier and standard basis diagonalized subspaces in finite abelian groups, revealing that small Fourier subspaces are generally distinct from standard subspaces, with implications for the Kadison-Singer problem.

Contribution

It provides a theoretical analysis of the discrepancy between Fourier and standard subspaces in finite abelian groups, connecting to recent solutions of the Kadison-Singer problem.

Findings

Small Fourier subspaces tend to be far from standard subspaces.

From within small Fourier subspaces, there exist standard subspaces nearly indistinguishable from their orthogonal complements.

The results relate to the structure of subspaces in the context of the Kadison-Singer problem.

Abstract

Let G be a finite abelian group. We examine the discrepancy between subspaces of l^2(G) which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier subspace whose dimension is small compared to |G| = dim(l^2(G)) tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison-Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is standard subspace which is essentially indistinguishable from its orthogonal complement.