Let $\mathcal{G}\subset L^2(\mathbb{R})$ be the subspace spanned by a Gabor Riesz sequence $(g,\mathrm{\Lambda })$ with $g\in L^2(\mathbb{R})$ and a lattice $\mathrm{\Lambda }\subset {\mathbb{R}}^2$ of rational density. It was shown recently that if g is well-localized both in time and frequency, then $\mathcal{G}$ cannot contain any time-frequency shift $\pi (z)g$ of g with $z\in {\mathbb{R}}^2\setminus \mathrm{\Lambda }$. In this paper, we improve the result to the quantitative statement that the $L^2$-distance of $\pi (z)g$ to the space $\mathcal{G}$ is equivalent to the Euclidean distance of z to the lattice Λ, in the sense that the ratio between those two distances is uniformly bounded above and below by positive constants. On the way, we prove several results of independent interest, one of them being closely related to the so-called weak Balian-Low theorem for subspaces.

让 $\mathcal{G}\subset {升}^2(\mathbb{电阻})$ 是 Gabor Riesz 序列跨越的子空间 $(G,\mathrm{\Lambda })$ 和 $G\in {升}^2(\mathbb{电阻})$ 和一个格子 $\mathrm{\Lambda }\subset {\mathbb{电阻}}^2$合理密度。最近的研究表明，如果g在时间和频率上都很好地定位，那么$\mathcal{G}$ 不能包含任何时频偏移 $\pi (z)G$的g与$z\in {\mathbb{电阻}}^2\setminus \mathrm{\Lambda }$. 在本文中，我们将结果改进为定量陈述，即${升}^2$-距离 $\pi (z)G$ 到空间 $\mathcal{G}$等价于z到晶格 Λ的欧几里得距离，这意味着这两个距离之间的比率由正常数上下统一限定。在此过程中，我们证明了几个独立感兴趣的结果，其中一个与子空间的所谓弱 Balian-Low 定理密切相关。