The HRT conjecture for time-frequency translates

Let $g\in L^2(\mathbb R,\mathrm{d}x)\setminus\{0\}$ and let $\{(a_k,b_k)\}_{k=1}^N$ be distinct points in $\mathbb R^2$. Define

$$g_k(x)=e^{2\pi i b_kx}g(x-a_k),\qquad 1\le k\le N.$$

HRT conjecture. The functions $g_k$, $1\le k\le N$, are linearly independent in $L^2(\mathbb R,\mathrm{d}x)$. This is a central open problem in time-frequency analysis, also called the Linear Independence Conjecture. Its Fock-space formulation asserts that distinct translates $U_{\lambda_k}f$ of every nonzero $f\in F^2$ are linearly independent; the paper uses a known special case to prove affirmative answers to several deep zero problems.