The abc-Problem for Gabor Systems

A longstanding problem in Gabor theory is to identify time-frequency shifting lattices $a/mathbb{Z}/times b/mathbb{Z}$ and ideal window functions $/chi_I$ on intervals $I$ of length $c$ such that $/{e^{-2/pi i n bt} /chi_I(t- m a):/ (m, n)/in /mathbb{Z}/times /mathbb{Z}/}$ 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 $abc$-problem for Gabor systems.