# A quantitative subspace Balian-Low theorem

## TL;DR

This paper establishes a quantitative version of the Balian-Low theorem for Gabor subspaces, showing a proportional relationship between the distance of time-frequency shifts and their proximity to the lattice, with implications for localization properties.

## Contribution

It provides a new quantitative bound relating the distance of time-frequency shifts to the subspace, extending the classical Balian-Low theorem to a more precise, measurable context.

## Key findings

- The $L^2$-distance of shifts is proportional to Euclidean distance to the lattice.
- The result applies to Gabor Riesz sequences with rational density lattices.
- Several auxiliary results related to the weak Balian-Low theorem are proved.

## Abstract

Let $\mathcal G\subset L^2(\mathbb R)$ be the subspace spanned by a Gabor Riesz sequence $(g,\Lambda)$ with $g\in L^2(\mathbb R)$ and a lattice $\Lambda\subset\mathbb R^2$ 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 $z\notin\Lambda$. 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 $\Lambda$, 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.