The $abc$-problem for Gabor systems

A Gabor system generated by a window function φ and a rectangular lattice aZ× Z/b is given by G(φ, aZ× Z/b) := {e−2πint/bφ(t−ma) : (m,n) ∈ Z× Z}. One of fundamental problems in Gabor analysis is to identify window functions φ and time-frequency shift lattices aZ × Z/b such that the corresponding Gabor system G(φ, aZ × Z/b) is a Gabor frame for L(R), the space of all square-integrable functions on the real line R. The range of density parameters a and b such that the Gabor system G(φ, aZ × Z/b) is a frame for L(R) is fully known surprisingly only for few window functions, including the Gaussian window and totally positive windows. Janssen’s tie suggests that the range of density parameters could be arbitrarily complicated for window functions, especially outside Feichtinger algebra. An eye-catching example of such a window function is the ideal window function on an interval. In this paper, we provide a full classification of triples (a, b, c) for which the Gabor system G(χI , aZ× Z/b) generated by the ideal window function χI on an interval I of length c is a Gabor frame for L(R). For the classification of such triples (a, b, c) (i.e., the abc-problem for Gabor systems), we introduce maximal invariant sets of some piecewise linear transformations and establish the equivalence between Gabor frame property and triviality of maximal invariant sets. We then study dynamic system associated with the piecewise linear transformations and explore various properties of their maximal invariant sets. By performing holes-removal surgery for maximal invariant sets to shrink and augmentation operation for a line with marks to expand, we finally parameterize those triples (a, b, c) for which maximal invariant sets are trivial. The novel techniques involving non-ergodicity of dynamical systems associated with some novel non-contractive and non-measure-preserving transformations lead to our arduous answer to the abc-problem for Gabor systems.