In 1996, C. Heil, J. Ramanatha, and P. Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any non-zero square integrable function $g$ and subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved.

In this talk, I will then introduce an inductive approach to investigate the conjecture, by attempting to answer the following question. Suppose the HRT conjecture is true for a function $g$ and a fixed set of $N$ points $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ For what other point $(a, b)\in \mathbb{R}^2\setminus \Lambda$ will the HRT remain true for the same function $g$ and the new set of $N+1$ points $\Lambda'=\Lambda \cup \{(a, b)\}$? I will illustrate this inductive argument on special classes of sets $\Lambda$ when $N\leq 4$. In addition, I will report on a recent joint work with V. Oussa in which we proved the conjecture for all configurations of the form $\Lambda=\{(a_k, b_k)\}_{k=1}^N \cup \{(a, b)\}$ where $\{(a_k, b_k)\}_{k=1}^N$ is either a subset of the unit lattice $\mathbb{Z}^2$ or all the points in this set lie on a line $L$.