  We investigate the properties of a sample of 35 galaxies, detected with ALMA
at 1.1 mm in the GOODS-ALMA field (area of 69 arcmin$^2$, resolution = 0.60",
RMS $\simeq$ 0.18 mJy beam$^{-1}$). Using the UV-to-radio deep multiwavelength
coverage of the GOODS-South field, we fit the spectral energy distributions of
these galaxies to derive their key physical properties. The galaxies detected
by ALMA are among the most massive at $z$ = 2-4 (M$_{\star,med}$ = 8.5$ \times$
10$^{10}$ M$_\odot$) and are either starburst or located in the upper part of
the galaxy star-forming main sequence. A significant portion of our galaxy
population ($\sim$ 40%), located at $z\sim$ 2.5-3, exhibits abnormally low gas
fractions. The sizes of these galaxies, measured with ALMA, are compatible with
the trend between $H$-band size and stellar mass observed for $z\sim2$
elliptical galaxies suggesting that they are building compact bulges. We show
that there is a strong link between star formation surface density (at 1.1 mm)
and gas depletion time: the more compact a galaxy's star-forming region is, the
shorter its lifetime will be (without gas replenishment). The identified
compact sources associated with relatively short depletion timescales
($\sim$100 Myr), are the ideal candidates to be the progenitors of compact
elliptical galaxies at $z$ $\sim$ 2.
