Math Conjectures

Heil–Ramanathan–Topiwala Conjecture

OPENMajorConjectureProposed 1996 · Full conjecture

Canonical statement

For every nonzero g∈L2(R)g\in L^2(\mathbb R) and every finite set of distinct points (aj,bj)∈R2(a_j,b_j)\in\mathbb R^2, the time--frequency translates
e2πibjxg(x−aj) e^{2\pi i b_jx}g(x-a_j)
are linearly independent in L2(R)L^2(\mathbb R).
View source LaTeX
For every nonzero \(g\in L^2(\mathbb R)\) and every finite set of distinct points \((a_j,b_j)\in\mathbb R^2\), the time--frequency translates
\[
  e^{2\pi i b_jx}g(x-a_j)
\]
are linearly independent in \(L^2(\mathbb R)\).
ANAL-FA-021:[HeilRamanathanTopiwala1996HRT][GuanOkoudjou2026HRT]
Analysis and functional analysis

Notes

The HRT conjecture says that finitely many distinct time-frequency shifts of a nonzero L2(R)L^2(\mathbb R) function are linearly independent [HeilRamanathanTopiwala1996HRT]. Many configurations and special windows are known, but the geometric freedom of a general finite set remains the obstruction. Work as recent as July 2026 still isolates unresolved four-point configurations [GuanOkoudjou2026HRT], confirming that the full standard statement remains open.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.