  We study diffraction corrections to the semiclassical spectral density of
dispersing (Sinai) billiards. They modify the contributions of periodic orbits
(PO's), with at least one segment which is almost tangent to the concave part
of the boundary. Given a wavenumber $k$, all the PO's with length up to the
Heisenberg length $O(k)$ are required for quantization. We show that most of
the contributions of PO's which are longer than a limit $O(k^{2/3})$ must be
corrected for diffraction effects. For orbits which just miss tangency, the
corrections are of the same magnitude as the semiclassical contributions
themselves. Orbits which bounce at extreme forward angles give very small terms
in the standard semiclassical theory. The diffraction corrections increase
their amplitude substantially.
