Linear independence of finite time-frequency shifts: the HRT conjecture

The HRT conjecture asks whether, for every nonzero f∈L²(R), every finite family of distinct time-frequency shifts x↦e^(2πibx)f(x−a) is linearly independent. The shift parameters (a,b) are distinct points of R².

Why it matters and what counts as progress

Why it matters: It asks for a universal linear-independence principle for finite time-frequency systems, beyond lattice or special-configuration results.

What counts as progress: A proof for a new family of finite shift configurations, or an explicit nonzero function with a finite dependent family. A counterexample must use distinct shifts.

How the problem developed

September 1996 · Conjecture [1]
Christopher Heil, Jayakumar Ramanathan and Pankaj Topiwala Posed the conjecture that every finite collection of time-frequency shifts of a nonzero L² function is linearly independent.
8 Jan 2025 · Partial result [2]
Jordy Timo van Velthoven Proved the conjecture for finite subsets of arbitrary discrete subgroups of phase space.
18 Aug 2026 · Counterexample [3]
Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven and Felix Voigtlaender Constructed a dependent system of 12 time-frequency shifts of a Schwartz function, disproving the conjecture in its unrestricted form.

AI contribution

Linear dependence of time-frequency shifts of a Schwartz function

Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven, Felix Voigtlaender

A nonzero Schwartz function has twelve distinct time-frequency shifts, eleven in one irrational translate of a half-integer lattice plus the origin, that are linearly dependent. The authors also prove a finite counterexample without a fixed twelve-shift count.

AI involvementAI-centered

ImportanceMajor

Technical verificationFormal (certified interval computation with python-flint/FLINT-Arb; exact rational bounds + human)

Human understandingFull