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HRT conjecture for finite Gabor systems

Let Λ={(xk,yk):1≤k≤N}⊂Rd×Rd\Lambda=\{(x_k,y_k):1\leq k\leq N\}\subset \mathbb{R}^d\times\mathbb{R}^d be a finite set of distinct points, and let 0≠f∈L2(Rd)0\neq f\in L^2(\mathbb{R}^d). HRT conjecture. The finite Gabor system

G(f,Λ):={MykTxkf:1≤k≤N}\mathcal{G}(f,\Lambda):=\{M_{y_k}T_{x_k}f:1\leq k\leq N\}

is linearly independent. The conjecture, formulated by Heil, Ramanathan, and Topiwala in 1996, is a central problem in Gabor analysis and concerns linear independence of finite sets of distinct time-frequency translates of a nonzero function. Its general status remains open.

Sources & referencesView supporting material

Primary source

Vignon Oussa, “Trichotomy for the HRT Conjecture for mixed integer configuration”, arXiv:2508.04613 (2026).

gabor-analysisharmonic-analysistime-frequency-analysis

Progress summary

Refreshed
Claimed solved

An August 2026 preprint claims a counterexample in one dimension, but the claim has not yet been independently verified.

Heil, Ramanathan, and Topiwala formulated the conjecture in 1996: every finite collection of distinct time-frequency shifts of a nonzero function should be linearly independent. The claim concerns arbitrary finite configurations and remains unsettled pending assessment of the new counterexample.

Known results

  • Independence holds for compactly supported, half-line-supported, and several rapidly decaying or analytic classes of generators.
  • It holds for configurations of at most three points and for various lattice, collinear, and four-point configurations.
  • A 2023 result covers nonzero meromorphic generators of a specified type.
  • A September 2025 result covers nonzero ultimately positive generators.

August 2026 claimed counterexample

An arXiv paper claims a nonzero Schwartz function f∗∈S(R)f_{*}\in\mathcal{S}(\mathbb{R}) with a linear dependence among twelve distinct shifts, disproving the conjecture already for d=1d=1. No independent verification, referee assessment, or corroborating exposition is supplied in the scan.

Current status (as of August 2026): A purported twelve-shift counterexample would settle the conjecture negatively, but it remains unverified; the general problem is therefore open, alongside established special cases.

Sources

Solutions 0

No solutions have been posted yet.