A minimal HRT counterexample at the sharp Gelfand--Shilov threshold

Abstract

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil--Ramanathan--Topiwala conjecture by constructing twelve linearly dependent time--frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand--Shilov class $\mathcal S_1^1(\mathbb R)$; their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in $\mathcal S_1^1(\mathbb R)$. Moreover, if $0\ne f\in\mathcal S_s^σ(\mathbb R)$ and $s<1$ or $σ<1$, then every finite system of distinct time--frequency shifts of $f$ is linearly independent. Thus four is the least possible cardinality and $(1,1)$ is the sharp coordinatewise threshold within the Roumieu Gelfand--Shilov scale.