  We show the full structure of the frame set for the Gabor system
$\mathcal{G}(g;\alpha,\beta):=\{e^{-2\pi i
m\beta\cdot}g(\cdot-n\alpha):m,n\in\Bbb Z\}$ with the window being the Haar
function $g=-\chi_{[-1/2,0)}+\chi_{[0,1/2)}$. The strategy of this paper is to
introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle,
and to provide a complete characterization of structures for its (symmetric)
maximal invariant sets. This transformation is related to the famous three gap
theorem of Steinhaus which may be of independent interest. Furthermore, a
classical criterion on Gabor frames is improved, which allows us to establish
{a} necessary and sufficient condition for the Gabor system
$\mathcal{G}(g;\alpha,\beta)$ to be a frame, i.e., the symmetric invariant set
of the transformation $\mathcal{M}$ is empty.
  Compared with the previous studies, the present paper provides a
self-contained environment to study Gabor frames by a new perspective, which
includes that the techniques developed here are new and all the proofs could be
understood thoroughly by the readers without reference to the known results in
the previous literature.
