Gelfand--Shilov 尖锐阈值处的一个最小 HRT 反例
A minimal HRT counterexample at the sharp Gelfand--Shilov threshold
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中文总结 AI 辅助
本文证明在 Roumieu Gelfand--Shilov 类 $\mathcal S_1^1(\mathbb R)$ 中可同时实现四点 HRT 反例与端点正则性，且四为最小基数，$(1,1)$ 为尖锐阈值。
中文摘要 AI 辅助
在一项引人注目的突破中，Faulhuber、Petersen、van Velthoven 和 Voigtlaender 通过构造一个 Schwartz 函数的十二个线性相关的时频平移，否证了 Heil--Ramanathan--Topiwala 猜想。次日，作者将其向量-Zak 与上同调架构压缩为一个显式的内在次临界四点构型，从而达到了最小可能的基数。不久之后，Jasper 和 Mixon 给出了该突破机制的透明解析改进，并证明对于每个丢番图对，HRT 反例可以用 Roumieu Gelfand--Shilov 类 $\mathcal S_1^1(\mathbb R)$ 中的窗函数实现；他们的定理在广泛的算术一般性下确立了端点正则性现象，同时不限制支撑基数。本文的目的是证明这两个尖锐进展可以同时实现。我们证明相同的显式四点、系数和矩阵余循环在 $\mathcal S_1^1(\mathbb R)$ 中允许一个非零窗函数。此外，若 $0\ne f\in\mathcal S_s^\sigma(\mathbb R)$ 且 $s<1$ 或 $\sigma<1$，则 $f$ 的每个有限个不同时频平移系统都是线性无关的。因此，四是最小可能的基数，且 $(1,1)$ 是 Roumieu Gelfand--Shilov 尺度内的尖锐坐标阈值。
英文摘要
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.
发表机构
- Bridgewater State University（布里奇沃特州立大学）
机构由 AI 辅助整理，请以论文原文为准。
