Proof of the HRT conjecture for configurations

We prove a particular case of the so-called “HRT conjecture”. More precisely, we show that given any planar trapezoid with vertices ( t j , ξ j ) j = 1 4 ∈ R 2 and given any nontrivial L 2 ( R ) function f, there is no linear dependence between the time-frequency translates f ( x + t j ) e 2 π i ξ j x . Our methods are mostly number theoretical.