Linear independence of coherent systems associated to discrete subgroups

TL;DR

This paper proves the linear independence of certain coherent systems linked to discrete subgroups, especially in nilpotent Lie groups and Euclidean space, providing new proofs for the HRT conjecture.

Contribution

It introduces a simple argument showing linear independence based on the absence of zero divisors in twisted group rings, applicable to discrete subgroups in nilpotent Lie groups.

Findings

Coherent systems are linearly independent when the twisted group ring has no zero divisors.

Verification of the zero divisor condition for discrete subgroups in nilpotent Lie groups.

A new, simple proof of the HRT conjecture for subsets of arbitrary discrete subgroups.

Abstract

This note considers the finite linear independence of coherent systems associated to discrete subgroups. We show by simple arguments that such coherent systems of amenable groups are linearly independent whenever the associated twisted group ring does not contain any nontrivial zero divisors. We verify the latter for discrete subgroups in nilpotent Lie groups. For the particular case of time-frequency translates of Euclidean space, our approach provides a simple and self-contained proof of the Heil--Ramanathan--Topiwala (HRT) conjecture for subsets of arbitrary discrete subgroups.