  We present an analysis of the spatial distribution of gas and galaxies using
new X-Shooter observations of $z\sim3-4$ quasars. Adding the X-Shooter data to
an existing dataset of high resolution quasar spectroscopy, we use a total
sample of 29 quasars alongside $\sim1700$ Lyman Break Galaxies in the redshift
range $2<z<3.5$. Analysing the Ly$\alpha$ forest auto-correlation function
using the full quasar sample, we find $s_0=0.081\pm0.006h^{-1}$Mpc. We then
investigate the clustering and dynamics of Ly$\alpha$ forest absorbers around
$z\sim3$ LBGs. From the redshift-space cross-correlation, we find
$s_0=0.27\pm0.14h^{-1}$Mpc, with power-law slope $\gamma=1.1\pm0.2$. We make a
first analysis of the dependence of this clustering length on absorber strength
based on cuts in the sightline transmitted flux, finding a clear preference for
stronger absorption features to be more strongly clustered around the galaxy
population than weaker absorption features. Further, we calculate the projected
correlation function, finding $r_0=0.24\pm0.04h^{-1}$Mpc (assuming a fixed
slope $\gamma=1.1$). Taking this as the underlying real-space clustering, we
fit the 2D cross-correlation function with a dynamical model incorporating the
infall parameter and the peculiar velocity, finding $\beta_{\rm F}=1.02\pm0.22$
and $240\pm60$ km s$^{-1}$ respectively. This result shows a significant
detection of gas infall relative to the galaxy population, whilst the measured
velocity dispersion is consistent with the velocity uncertainties on the galaxy
redshifts. We evaluate the Cauchy-Schwarz inequality between the galaxy-galaxy,
absorber-absorber, and galaxy-absorber correlation functions, finding a result
significantly less than unity: $\xi_{\rm ag}^2/(\xi_{\rm gg}\xi_{\rm
aa})=0.25\pm0.14$, implying that galaxies and Ly$\alpha$ absorbers do not
linearly trace the underlying dark matter distribution in the same way.
