标题： HRT猜想的混合整数配置的三重性 显示英文标题

标题： Trichotomy for the HRT Conjecture for mixed integer configuration

摘要： 设$\Lambda$由$N-1$个格点在$\mathbb{Z}^d \times \mathbb{Z}^d$上以及一个非格点点$(\alpha,\beta)$组成。 我们证明，对于每个非零窗口$f \in \mathcal{S}(\mathbb{R}^d)$，有限的Gabor系统${M_y T_x f : (x,y) \in \Lambda}$是线性无关的。 一个假定的依赖关系通过Zak变换产生一个由环面平移$z \mapsto z + (-\alpha,\beta)$驱动的模和相位方程组。 将所得轨道分类为稠密、有限或无限非稠密，得到一个严格的三元划分：(i) 稠密轨道迫使$f=0$，(ii) 有限轨道归约为 Linnell 的格定理，(iii) 一个新的刚性论证排除了无限非稠密的情况。 显示英文摘要

摘要： Let $\Lambda$ consist of $N-1$ lattice points in $\mathbb{Z}^d \times \mathbb{Z}^d$ together with a single off-lattice point $(\alpha,\beta)$. We prove that the finite Gabor system ${M_y T_x f : (x,y) \in \Lambda}$ is linearly independent for every non-zero window $f \in \mathcal{S}(\mathbb{R}^d)$. A supposed dependence gives rise, via the Zak transform, to a system of modulus and phase equations driven by the torus translation $z \mapsto z + (-\alpha,\beta)$. Classifying the resulting orbits as dense, finite, or infinite-non-dense yields a sharp trichotomy: (i) dense orbits force $f=0$, (ii) finite orbits reduce to Linnell's lattice theorem, and (iii) a new rigidity argument excludes the infinite non-dense case.