  We present an Atacama Large Millimeter/submillimeter Array (ALMA) survey of
CO(4--3) line emitting galaxies in 17 quasar fields at $z\sim4$ aimed to
perform the first systematic search of dusty galaxies in high$-z$ quasar
environments. Our blind search of galaxies around the quasars results in 5 CO
emitters with S/N$\geq5.6$ within a projected radius of
$R\lesssim1.5\,h^{-1}\,$cMpc and a velocity range of
$\rm\Delta\,v=\pm1000\,$km\,s$^{-1}$ around the quasar. In blank fields, we
expect to detect only 0.28 CO emitters within the same volume, implying a total
overdensity of $17.6^{+11.9}_{-7.6}$ in our fields, and indicating that quasars
trace massive structures in the early universe. We quantify this overdensity by
measuring the small-scale clustering of CO emitters around quasars, resulting
in a cross-correlation length of $r_{\rm
0,QG}=8.37^{+2.42}_{-2.04}\,h^{-1}\,$cMpc, assuming a fixed slope $\gamma=1.8$.
This contradicts the reported mild overdensities (x1.4) of Lyman alpha emitters
(LAEs) in the same fields at scales of $R\lesssim7\,h^{-1}\,$cMpc which is well
described by a cross-correlation length 3 times lower than that measured for CO
emitters. We discuss some possibilities to explain this discrepancy, including
low star formation efficiency, and excess of dust in galaxies around quasars.
Finally, we constrain, for the first time, the clustering of CO emitters at
$z\sim4$, finding an auto-correlation length of $r_{\rm
0,CO}=3.14\pm1.71\,h^{-1}\,$cMpc (with $\gamma=1.8$). Our work together with
the previous study of LAEs around quasars traces simultaneously the clustering
properties of both optical and dusty galaxy populations in quasars fields,
stressing the importance of multi-wavelength studies, and highlighting
important questions about galaxy properties in high$-z$ dense environments.
