Linear Independence of Four Time-Frequency Shifts

Definitions and mathematical statement

Work in the complex Hilbert space $L^2({\mathbb{R}},{\,\mathrm{d}} x)$, where functions are identified almost everywhere. For $a,b\in{\mathbb{R}}$, define translation and modulation by \[ (T_ag)(x)=g(x-a),\qquad (M_bg)(x)=e^{2\pi i b x}g(x). \] The exact source question is whether, for every $g\in L^2({\mathbb{R}})$ with $\|g\|_2>0$, the family \[ \{g,T_1g,T_2g,M_1g\} \] is linearly independent over ${\mathbb{C}}$. Equivalently, does \[ c_0g(x)+c_1g(x-1)+c_2g(x-2)+c_3e^{2\pi i x}g(x)=0 \quad\text{for almost every }x\in{\mathbb{R}} \] with $(c_0,c_1,c_2,c_3)\in{\mathbb{C}}^4$ imply $c_0=c_1=c_2=c_3=0$, for every such $g$? The four time-frequency parameters are exactly $(0,0),(1,0),(2,0),(0,1)$; they are not arbitrary four distinct points.

The supplied literature triage, checked 2026-08-21, says that the general four-point HRT problem was unresolved and additionally claims an August 2026 twelve-shift counterexample to the unrestricted HRT conjecture, linking two preprints. Those are supplied, time-sensitive literature assertions requiring independent verification, particularly the claimed counterexample. Neither an unresolved general four-point problem nor a proposed twelve-shift example determines the status of this specific integer-parameter configuration. The registry's open label is therefore recorded as curation metadata rather than certified here.

Short English statement

For every nonzero square-integrable function on ${\mathbb{R}}$, ask whether its three integer translates $g,T_1g,T_2g$, together with its unit modulation $M_1g$, are linearly independent. The fixed configuration must be distinguished from the general four-point HRT question and from the supplied unverified twelve-shift claim.

Sources

• Local UnsolvedMath registry, permanent ID 30000037, catalogue entry, Linear Independence of Four Time-Frequency Shifts.
• Mini-Workshop: Wavelets and Frames (2004), Oberwolfach Reports, pp. 479--544, DOI https://doi.org/10.4171/owr/2004/10.
• Supplied reference labeled "Four-point HRT results": https://arxiv.org/abs/2607.26878.
• Supplied reference labeled "HRT counterexample": https://arxiv.org/abs/2608.05044. Its asserted counterexample is not independently verified in this draft.
• Supplied literature triage dated 2026-08-21; no author lists, titles, theorem numbers, or verification details for the two preprints are included.