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We present an analysis of the spatial distribution of gas and galaxies usingnew X-Shooter observations of $z\sim3-4$ quasars. Adding the X-Shooter data toan existing dataset of high resolution quasar spectroscopy, we use a totalsample of 29 quasars alongside $\sim1700$ Lyman Break Galaxies in the redshiftrange $2<z<3.5$. Analysing the Ly$\alpha$ forest auto-correlation functionusing the full quasar sample, we find $s_0=0.081\pm0.006h^{-1}$Mpc. We theninvestigate 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 afirst analysis of the dependence of this clustering length on absorber strengthbased on cuts in the sightline transmitted flux, finding a clear preference forstronger absorption features to be more strongly clustered around the galaxypopulation than weaker absorption features. Further, we calculate the projectedcorrelation function, finding $r_0=0.24\pm0.04h^{-1}$Mpc (assuming a fixedslope $\gamma=1.1$). Taking this as the underlying real-space clustering, wefit the 2D cross-correlation function with a dynamical model incorporating theinfall parameter and the peculiar velocity, finding $\beta_{\rm F}=1.02\pm0.22$and $240\pm60$ km s$^{-1}$ respectively. This result shows a significantdetection of gas infall relative to the galaxy population, whilst the measuredvelocity dispersion is consistent with the velocity uncertainties on the galaxyredshifts. We evaluate the Cauchy-Schwarz inequality between the galaxy-galaxy,absorber-absorber, and galaxy-absorber correlation functions, finding a resultsignificantly less than unity: $\xi_{\rm ag}^2/(\xi_{\rm gg}\xi_{\rmaa})=0.25\pm0.14$, implying that galaxies and Ly$\alpha$ absorbers do notlinearly trace the underlying dark matter distribution in the same way.

The VLT LBG Redshift Survey - VI. Mapping HI in the proximity of $z\sim3$ LBGs with X-Shooter