The HRT Conjecture for Symmetric Configurations and Real-Valued Functions

The Heil-Ramanathan-Topiwala (HRT) conjecture asserts that every finite collection of distinct time-frequency shifts of a nonzero square-integrable function is linearly independent. Despite its simple formulation, the conjecture remains open even under strong regularity and decay assumptions on the generating function, and in particular for general configurations of four distinct points. In this paper, we establish the HRT conjecture for an infinite family of symmetric $(2n+1,2)$ configurations and arbitrary functions in $L^2(\mathbb{R})$. More generally, our argument applies whenever the collinear points have commensurable spacings. As a consequence, we prove the HRT conjecture for every configuration of four distinct points when the generating function is real-valued. The proof combines a reduction to products of trigonometric polynomials with estimates along orbits of irrational rotations.