Deep zero problems and the HRT conjecture
Yufei Li, Zeguang Liu, Kehe Zhu
TL;DR
This work connects deep zero problems in the Fock space to the HRT conjecture in time-frequency analysis by leveraging Weyl unitaries and a $d$-fold decomposition of analytic functions. It proves conditional deep zero results: if a finite-N HRT-type conjecture holds for $N=d$, then mixed vanishing conditions imply $f=0$, yielding affirmative answers for $d=2$ and $d=4$, and new results for $d=3$ and $d=6$. The paper also confirms HRT-type results for root-of-unity configurations with $d\in\{1,2,3,4,6\}$ and shows that for other $d$ these roots of unity generally do not lie on a regular lattice, motivating the deep zero framework. The discussion culminates in an explicit open problem on whether the HRT conjecture holds for $oldsymbol{ u}_k=\beta e^{2k\pi i/d}$ when $d=5$ or larger, guiding future investigations at the interface of complex analysis and time-frequency analysis.
Abstract
We investigate a "deep zero problem" proposed by Hedenmalm. We show that there is a natural connection between Hedenmalm's problem and the classical HRT conjecture in time-frequency analysis. This connection allows us to show that Hedenmalm's problem 5.2 in [5] as well as some of its natural analogs have affirmative answers.