Given a non-zero square integrable function $$g$$ and $$\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2$$ let \( \mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.\) The Heil-Ramanathan-Topiwala (HRT) Conjecture is the question of whether $$\mathcal{G}(g, \Lambda)$$ is linearly independent. For the last two decades, very little progress has been made in settling the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture. I will then describe some recent attempts in settling the conjecture for some special classes of functions.

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