Syndetic Sets, Paving, and the Feichtinger Conjecture

TL;DR

This paper establishes that certain partitions of Bessel sequences and Fourier frames into Riesz basic sequences can be chosen to be syndetic, linking harmonic analysis with group theory and von Neumann algebras.

Contribution

It proves that partitions of invariant Bessel sequences and Fourier frames into Riesz basic sequences can be made syndetic, connecting frame theory with group and operator algebra structures.

Findings

Partitioned sequences can be chosen to be syndetic.

Results apply to Fourier frames in higher dimensions.

Links between frame partitions and von Neumann algebra properties.

Abstract

We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen to be syndetic. We then apply this result to prove that if a Fourier frame for a measurable subset of a higher dimensional cube can be partitioned into Riesz basic sequences, then each subset can be chosen to be a syndetic subset of the corresponding higher dimensional integer lattice. Both of these results follow from a result about syndetic pavings of elements of the von Neumann algebra of a discrete group.