  The structural properties of static, jammed packings of monodisperse spheres
in the vicinity of the jamming transition are investigated using large-scale
computer simulations. At small wavenumber $k$, we argue that the anomalous
behavior in the static structure factor, $S(k) \sim k$, is consequential of an
excess of low-frequency, collective excitations seen in the vibrational
spectrum. This anomalous feature becomes more pronounced closest to the jamming
transition, such that $S(0) \to 0$ at the transition point. We introduce an
appropriate dispersion relation that accounts for these phenomena that leads us
to relate these structural features to characteristic length scales associated
with the low-frequency vibrational modes of these systems. When the particles
are frictional, this anomalous behavior is suppressed providing yet more
evidence that jamming transitions of frictional spheres lie at lower packing
fractions that that for frictionless spheres. These results suggest that the
mechanical properties of jammed and glassy media may therefore be inferred from
measurements of both the static and dynamical structure factors.
