The Heil-Ramanathan-Topiwala (HRT) conjecture states that no non-zero function can satisfy a finite linear relation between its time-frequency shifts under certain decay conditions. While many positive results existed—such as those by Linnell for discrete subgroups or Bownik and Speegle for super-exponential decay—the conjecture remained open for the Schwartz class.
Recently, Faulhuber, Petersen, van Velthoven, and Voigtlaender resolved this by providing a counterexample. They proved that there exist complex numbers, distinct points, and a non-zero Schwartz function that satisfy the relation. Notably, this breakthrough was AI-assisted, with the authors using artificial intelligence to develop the initial proof strategy and numerical verification.
The researchers utilized a vector-valued version of the Zak transform to transform the problem into a vector cocycle equation. By using numerical computation and AI-assisted guesswork, they approximated a rank-one function, allowing them to apply a contraction mapping argument. This approach successfully navigated the complexities of the eigenvalue problem, effectively resolving a long-standing problem in time-frequency analysis.
Source: Hacker News · Summarized by HeadlinesBriefing