A longstanding problem in Gabor theory is to identify time-frequency shifting lattices $amathbb imes bmathbb$ and ideal window functions $chi_I$ on intervals $I$ of length $c$ such that ${e^{-2pi i n bt} chi_I(t- m a): (m, n)in mathbb imes mathbb}$ are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above...