The strong HRT conjecture asserts that the time-frequency
translates of any nontrivial function in $L^2(\mathbb R)$ are linearly
independent. The weak HRT conjecture has the same formulation, but this time
for Schwartz functions. Prior to our work, the only result of a reasonably
general nature was Linnell's proof in the case when the translates belong to
a lattice.
I will first describe an alternative argument to Linnell's (joint work with
Zubin Gautam), inspired by the theory of random Schr\"odinger operators.
Then I will explore both some solo and joint work (with Zaharescu) involving
a number theoretical approach to the HRT conjecture, for some special 4
point configurations.