The ALMA Spectroscopic Survey in the HUDF: Deep 1.2 mm continuum number
  counts
  We present the results from the 1.2 mm continuum image obtained as part of
the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS). The 1.2
mm continuum image has a size of 2.9 (4.2) arcmin$^2$ within a primary beam
response of 50% (10%) and a rms value of $9.3\thinspace{\rm\mu Jy\thinspace
beam^{-1}}$. We detect 35 sources at high significance (Fidelity $\geq0.5$), 32
of these have well characterized near-infrared HST counterparts.
  We estimate the 1.2 mm number counts to flux levels of $<30\thinspace{\rm\mu
Jy}$ in two different ways: we first use the detected sources to constrain the
number counts and find a significant flattening of the counts below $S_\nu \sim
0.1$ mJy. In a second approach, we constrain the number counts by using a
probability of deflection statistics (P(D)) analysis. For this latter approach,
we describe new methods to accurately measure the noise in interferometric
imaging (employing jack-knifing in the cube and in the visibility plane). This
independent measurement confirms the flattening of the number counts. Our
analysis of the differential number counts shows that we are detecting
$\sim$93% ($\sim$100% if we include the lower fidelity detections) of the total
continuum dust emission associated to galaxies in the HUDF.
  The ancillary data allows us to study the dependence of the 1.2 mm number
counts on redshift ($z=0-4$), galaxy dust mass (${\rm M}_{\rm
dust}=10^{7}-10^{9}{\rm M}_{\odot}$), stellar mass (${\rm
M}_{*}=10^{9}-10^{12}{\rm M}_{\odot}$), and star-formation rate (${\rm
SFR}=1-1000\thinspace{\rm M}_{\odot}\thinspace{\rm yr^{-1}}$). In an
accompanying paper we show that the number counts are crucial to constrain
galaxy evolution models and the understanding of star-forming galaxies at high
redshift.
