When a characteristic function generates a Gabor frame

We investigate the characterization problem which asks for a classification of all the triples (a,b,c) such that the Gabor system is a frame for L2(R). We present a new approach to this problem. With the help of a set-valued mapping defined on certain union of intervals, we are able to provide a complete solution for the case of ab being a rational number. For the irrational case, we prove that the classification problem can also be completely settled if the union of some intervals obtained from the set-valued mapping becomes stabilized after finitely many times of iterations, which we conjecture is always true.